Thermal Hall conductivity and topological transition in a chiral p-wave superconductor for Sr2RuO4

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by  Yoshiki Imai

Thermal Hall conductivity and topological transition in a chiral p-wave
superconductor for Sr RuO
2 4
Yoshiki Imai1,∗ Katsunori Wakabayashi2,3, and Manfred Sigrist4
1Department of Physics, Saitama University, Saitama 338-8570, Japan
2International Center for Materials Nanoarchitectonics (WPI-MANA),
National Institute for Materials Science (NIMS), Tsukuba 305-0044, Japan
3Kwansei Gakuin University, Sanda 669-1337, Japan and
6 4Theoretische Physik, ETH-Zu¨rich, CH-8093 Zu¨rich, Switzerland
1 (Dated: January 15, 2016)
0
The interplay between the thermal transport property and the topological aspect is investigated
2
in a spin-triplet chiral p-wave superconductor Sr2RuO4 with the strong two-dimensionality. We
n show thethermal Hall conductivityis well described bythetemperature linear term and theexpo-
a nential term in the low temperature region. While the former term is proportional to the so-called
J
Chern number directly, the latter is associated with the superconducting gap amplitude of the γ

4 band. We also demonstrate that the coefficient of the exponential term changes the sign around
1 Lifshitz transition. Ourobtained result may enable usaccess easily thephysical quantitiesand the
topological property of Sr2RuO4 in detail.
] PACS numbers:
n
o
c
- I. INTRODUCTION cell [13, 14]. We have pointed out recently that such a
r rotation may change the Fermi surface topology of the
p
u The transition metal oxide superconductor γ band pushing the Fermi energy through the van Hove
s Sr RuO [1–3] has attracted much interest as a points and in this way generate a Lifshitz transition be-
t. str2ong c4andidate for topological superconductivity. The tween an electron- and hole-like shape [15]. In addition,
a
m superconducting state is characterized by spin-triplet the studies of the electron doping effect of Sr2RuO4 by
Cooper pairing [4] and broken time-reversal symme- La substitution for Sr ions [16–18], the uniaxial pressure
d- try [5,6]. Within the traditionalsymmetry classification and the strain effects on Sr2RuO4 [19–21] show that the
n scheme the order parameter with chiral p-wave Cooper topology of the γ band changes around the van Hove
o pairing remains as the only phase compatible with these points in the whole bulk system. In the chiral supercon-
c experiments: d(k) = ∆ zˆ(k ±ik ) characterized by an ductingphasetopologicalpropertydependsonthestruc-
[ orbitalangularmoment0um Lxz =±y1 alongthe z-axis[7]. ture of the γ band [12], and the different Fermi surface
1 This isananalogtothe A-phase ofsuperfluid3Hewhich topologies give different Chern numbers.
v in a quasi-two-dimensional system as Sr RuO opens a Ithasbeenshownthatthechiralp-wavesuperconduc-
2 4
4 full quasiparticle excitation gap and whose topological tor supports chiral edge states responsible for sponta-
5 character can be labeled by Chern number. neous in-plane supercurrents along the surfaces [22, 23].
4
Angle-resolved photoemission-spectroscopy (ARPES) Thesecurrent,however,arenotadirectfeatureoftopol-
3
0 andde Haas-vanAlphen data [8–10]aswellasfirstprin- ogy, in the sense that their magnitude depends strongly
. ciples calculations reveal that the γ band, a genuinely on the band structure and the orientationand quality of
1
cylindrical (two-dimensional) electron-like Fermi surface the surface [15, 24, 25]. It has been shown that surface
0
andderivedfromtheRu4d-t d orbital,hasitsFermi currents are essentially insensitive to changes of Chern
6 2g xy
1 level only slightly below the van Hove singularity. This number, e.g. through a Lifshitz transition [15]. More-
: means that the Fermi surface approaches the Brillouin over, surface currents need not flow in a simple circular
v
zone boundary closely at the saddle points (π/a,0) and patternaroundadisk-shapedsampleasnaivelyexpected,
i
X (0,π/a) with a lattice constant a, such that Sr RuO but can even show peculiar current reversals [24]. Thus,
2 4
r could be close to a Lifshitz transition. In the super- surface supercurrents are not an optimal feature to test
a conducting phase the energy gap is suppressed strongly the topology of the superconducting phase.
by symmetry close to these van Hove points. Topology On the other hand, the thermal Hall effect (Righi-
induced features, therefore, are fragile against disorder Leduceffect)ismoresuitabletostudythetopologyofthe
effects and thermal broadening which destroy the quasi- superconducting phase, as it realizes quantization fea-
particle gap [11, 12]. tures which are directly connected with the Chern num-
At c-axis oriented surfaces of Sr RuO a lattice re- bers. In the following we will discuss the thermal Hall
2 4
constructionoccurs,wherebytheRuO octahedrarotate effect in the context of topology, using a lattice fermion
6
around their c-axis leading to the doubling of the unit modelwithanattractiveinter-siteinteractionwithinthe
BCS-type mean-field approximation. Moreover, we an-
alyze the temperature-dependence of the thermal Hall
conductivity, in particular, near the Lifshitz transition
∗ [email protected] changing the Chern number.
2
II. MODEL order parameter as follows,
1
The electronic properties of Sr2RuO4 are governed by ∆x = 2(hci↑cix↓i+hci↓cix↑i), (4)
the Ru 4d-t orbitals, and the electron bands yield the
2g 1
three Fermi surfaces, α, β and γ. The hole-like α and ∆ = (hc c i+hc c i), (5)
y 2 i↑ iy↓ i↓ iy↑
the electron-like β bands have one-dimensional charac-
1
ters, whose Fermi surface topologies are robust under a ∆ = (hc c i+hc c i), (6)
+ 2 i↑ i+↓ i↓ i+↑
change of the chemical potential and topologicalproper-
1
ties are opposite to eachother and, thus, vanish [26, 27]. ∆ = (hc c i+hc c i), (7)
Therefore,forsimplicity,wewillfocushereontheγ band − 2 i↑ i−↓ i↓ i−↑
which remains the only one essential for topological as-
wherec istheannihilationoperatorford -orbitalelec-
i↑ xy
pects.
tronsonthesiteiwithspinσ(↑or↓). Theindicesx,y,+
The left panel of Fig. 1 shows the lattice structure
and−standforthe(a,0),(0,a),(a,a)and(−a,a)direc-
used for our model, whereby t (t′) and U (V) denote
tions of real-space pairing, respectively. In the following
the hopping amplitudes and the coupling constants of we will suppress the constants a, ~ and k by setting
B
theattractiveinteractionbetweennearestneighbor(next them 1. The next nearest hopping amplitude t′ = 0.35t
nearestneighbor)sites. Thesetwoparametersprovidean
reproduces the γ band of Sr RuO .
2 4
enlargedrangeoftopologicalsuperconductingstateswith
We can tune the Lifshitz transition mentioned above
different Chern numbers.
directlybychangingthechemicalpotentialµ. Inthisway
Forthissingle-bandtight-bindingmodel,weintroduce
the Fermi surface switches from a electron- to hole-like
the following Bogoliubov-de Gennes (BdG) Hamilto- shape at a critical value of µ=µ =4t′ =1.4t shown in
c
nian yielding a spin-triplet pairing on a two-dimensional
Fig. 1 rightpanel. At the same time the topology of the
square lattice
superconducting phase changes as we will show below.
HBdG =Xk (cid:16)c†k↑,c−k↓(cid:17)(cid:18)∆εk∗k −∆εkk(cid:19)(cid:18)cc†−kk↑↓(cid:19), (1) III. RESULTS
wofhaenreecle†kcσtr(ocnkσw)itihs twhaevecrveeacttioonr k(a(n=nih(kila,tkion))) aonpdersaptoinr The order parameters ∆x,y and ∆± are determined
x y
self-consistently. Upon the convergence we obtain the
σ(=↑,↓). The normal state electron dispersion of the γ
chiral p-wave superconducting phase as the most sta-
band and the gap function are parametrized as
ble state, with ∆ = ±i∆ favored by U 6= 0 and
y x
εk =−2t(coskx+cosky)−4t′coskxcosky −µ, (2) ∆− =±i∆+ driven by V 6=0. Figure 2 shows the order
parameterswhereIm∆ forU 6=0andV =0(Im∆ for
∆k =2iU{∆xsinkx+∆ysinky} y −
U =0 and V 6=0) is identicalto Re∆ (Re∆ ) with the
x +
+2iV {∆ sin(k +k )+∆ sin(−k +k )},(3)
+ x y − x y orbital angular momentum L = +1. Note that the or-
z
derparametersbetweenthenearestandthenextnearest
whereµisthechemicalpotential. Moreover,thesymbols
neighbor sites always have same chirality, whose phase
∆ represent the spin-triplet superconducting
m(=x,y,+,−)
difference is arg(∆ /∆ )=π/4 when both U and V do
+ x
not vanish [15]. Hereafter we focus on the specific angu-
lar momentum L =+1 state. Thus the d vector can be
z
1 rewritten as d = ∆xzˆ(sinkx +isinky)+∆+zˆ{sin(kx +
U V V ky)+isin(−kx+ky)}.
0.8
t’ t’ U pk /y0.6 mm ==11..54tt (a) V =0 (b) U =0
0.4 m =1.3t
0.1 0.1
t t 0.2 x +
DRe UU==--t2t DRe VV==--t2t
0 0.2 0.4 0.6 0.8 1 0.05 0.05
kx /p
FIG. 1. (Color online) (Left panel) Lattice structure with 0 0
1.3 1.4 1.5 1.3 1.4 1.5
square lattice. t (t′) stands for the hopping amplitude be- m /t m /t
tween nearest (next nearest) neighbor lattice sites. U (V)
represents the attractive interaction between nearest (next FIG. 2. (Color online) The order parameters as a function
nearest) neighbor lattice sites. (Right panel) The Fermi sur- of chemical potential µ at zero temperature. The left (right)
faceforseveralchoicesofthechemicalpotentialµinthenor- panel represents Re∆x for U = −t, −2t with V = 0 (Re∆+
mal phase. for V =−t,−2t with U =0).
3
3 (a) V=0, m =1.3t (b) V=0, m =1.5t
U=-2t, V=0 0.01 0.01
er2 U=0, V=-2t
b
m
nu1 kxy 0 kxy 0
Chern 0 -0.01 UUU===---t12.t5t -0.01 UUU===---t12.t5t
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1
-1 (c) U=0, m =1.3t T/t (d) U=0, m =1.5t T/t
1.3 1.4 1.5 0.05 0.05
m /t
0.04 0.04
V=-t
0.03 0.03 V=-1.5t
FIG. 3. The Chern number as a function of chemical po- xy xy V=-2t
tential µ for (U,V) = (−2t,0) and (0,−2t) at absolute zero k0.02 k0.02
temperature. V=-t
0.01 V=-1.5t 0.01
V=-2t
0 0
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1
T/t T/t
The bulk quasiparticle spectrum is given by Ek± = FIG. 4. (Color online) The thermal Hall conductivity κxy as
± ε2k+|∆k|2 and at the van Hove point (k = (π,0)) a function of temperature at absolute zero temperature for
Ekp± = ±(4t′−µ), where the gap function ∆k vanishes. several choices of U with V = 0 ((a) and (b)) and several
Thereareseveralotherdiscretepointsofzerogapwithin choicesofV withU =0((c)and(d))atµ=1.3t(leftpanels)
the Brillouin zone. andµ=1.5t (rightpanels). Thesolid-red linerepresentsκLxy
for each set of the parameters (U, V, µ).
First we examine the Chern number N as a function
c
ofthechemicalpotentialµfortwochoicesofinteractions
(U,V) at zero temperature, as depicted in Fig. 3. While
for the electron-like Fermi surface the Chern number
takesthetwovalues−1and+3for(U,V)=(−2t,0)and This term linear in temperature is defined as
(U,V) = (0,−2t) at µ < µ , respectively, it is uniquely
c
+1 for µ > µc, the hole-like Fermi surface. This change κL ≡ πNcT. (10)
asfunctionofµdefinestheLifshitztransitionatµ . Note xy 12
c
µ isindependentofthemagnitudeofU andV. Thetwo
c
valuesofNc forµ<µc suggestthepresenceofatopolog- The change of the low-T slope of κxy (corresponding to
icaltransitionwhenwecontinuouslyinterpolatebetween κLxy)indicatesthechangeoftheChernnumberatµ=µc
the two cases of (U,V), yielding a change of Nc by 4. andconveysthatintheverylowtemperatureregimeκxy
We turn now to the thermal transport properties. It is uniquely determined by topological properties of the
has been noticed that for the ordinary thermal current superconducting phase.
the circulating contribution of the chiral edge states has We observe a deviation from the T-linear behavior of
be considered with care [28]. We will focus here, how- κ inFig. 4whichwetracebacktothefinitemagnitude
xy
ever,onthe thermalHall conductivity in a chiral p-wave of the quasiparticle excitation gap, as we will demon-
superconductor which can be expressed as strate in the following. For this purpose we discuss the
influence ofthe structureofΛ(ε)onthe temperaturede-
1
κ =− dεε2Λ(ε)f′(ε), (8) pendence of κxy. Figure 5 shows Λ(ε) for severalchoices
xy
4πT Z of (U,V) at µ = 1.3t and µ = 1.5t. In our numerical
4π ∂ukn ∂ukn resultswefindΛ(ε)=Nc inaregion|ε|≤ε0 andarapid
Λ(ε)= M Xk,nIm(cid:26)(cid:28) ∂kx (cid:12)(cid:12) ∂ky (cid:29)(cid:27)θ(ε−Ekn), (9) cthheanqgueabsiepyaorntidc.leBgyapcoEmp(atrhiseolnowweestcaenxccitoantnioencteεn0erwgiyt)h,
(cid:12) g
(cid:12) i.e. ε =E /2. For|ε|≫ε the valueofΛ(ε)shrinksto-
where f′(ε) is the derivative of the Fermi-distribution 0 g 0
wards zero. This part of the function depends on details
function[29],T andM denotetemperatureandthenum-
of the model.
ber of sites, respectively, and ukn (Ekn) is the periodic
For an illustrative approximation of κ as a function
part of the Bloch wave function (the eigenvalue) of the xy
BdG equations (Eq. (1)) for the wave vector k and oftemperatureweuseapiece-wiseconstant(boxshaped)
function Λ′(ε) instead of the exact Λ(ε),
band index n and is obtained numerically. Note that
Λ(0)(≡N ) is identical to the Chern number.
c
N |ε|≤ε
Figure 4 displays the temperature dependence of the Λ′(ε)= c 0 . (11)
thermalHall conductivity. In the low-temperaturelimit, (cid:26) 0 otherwise
the thermal Hall conductivity, κ ≈ (πN /12)T [29],
xy c
is proportional to temperature with a prefactor directly Then the evaluation of Eq. (8) is straightforward and
relatedtotheChernnumber(thesolid-redlinesinFig.4). showstheessentialbehaviorofapproximatethermalHall
4
conductivity κ′ with Λ′(ε), (a) (U, V )=(-t, 0) (b) (U, V )=(-2t, 0)
xy 0.15
0.04
κ′xy = N2cπT (cid:26)π62 −γ(T)e−εT0(cid:27), (12) 0.02 Nc=-1 Nc=+1 ed0 0.1 Nc=-1 Nc=+1
γ(T)≡4 ε0 2+4 ε0 +2. (13) 0 b 0.05 edb0
(cid:16)2T(cid:17) (cid:16)2T(cid:17) 0
-0.02
The second term in Eq. (12) is obviously a correction 1.3 1.4 1.5 1.3 1.4 1.5
m/t m/t
to κLxy due to contributions ofthermallyactivatedquasi- (c) (U, V )=(0, -t) (d) (U, V )=(0, -2t)
particles and is only a valid approximation as long as
ε0 ≫ T, also in view of the temperature dependence of 0.05 Nc=+3 Nc=+1 0.1 Nc=+3 Nc=+1
the gap which shrinks as temperature increases.
We now compare κ (T) with our numerical results.
xy
0 0
To simplify the expression we use here ed0 ed0
κxy ≈κ˜xy ≡κLxy+βe−Tδ, (14) 1.3 1.4 b1.5 1.3 1.4 b1.5
m/t m/t
with fitting parameters β and δ. Thus, we may use an
Arrhenius fit for κ −κL . Figure 6 displays the fitting FIG. 6. (Color online) The fitting parameters δ and β as a
xy xy
parameters δ and β for several choices of (U,V). In function of chemical potential µ for (U,V)=(−t,0),(−2t,0)
the whole range of µ, δ is essentially identical with ε , (upper panels) and (U,V) = (0,−t),(0,−2t) (lower panels).
0
whereε ≈|4t′−µ|fortheFermisurfaceclosetothevan The solid lines stand for thehalf energy gap ǫ0 in all panels.
0
Hovepoint. Thus,approachingthe (topological)Lifshitz
transition the quantization becomes thermally softened
(a) (U, V )=(-t, 0) (b) (U, V )=(-2t, 0)
as the protection due to the quasiparticle gap weakens. 0.01
Thecomparisonofκxywithκ˜xyfortheestimatedδand 0.01 m=1.3t
β is depicted in Fig. 7 for µ = 1.3t and µ = 1.5t. The m=1.5t
approximation works well in the temperature range of xy 0 xy 0
k k
validity(ǫ ≫T). Theparameterβ includesinformation
0
of the shape to Λ(ε) and determines, in particular, the m=1.3t -0.01
m=1.5t
sign of the deviation from T-linear behavior. We find -0.01
0 0.01 0.02 0.03 0.04 0 0.02 0.04 0.06 0.08 0.1
that β has the same sign as the Chern number Nc, if T/t T/t
(c) (U, V )=(0, -t) (d) (U, V )=(0, -2t)
Λ(0) = N represents a global maximum of Λ(ε). It is 0.015 0.04
c
opposite, if Λ(0)=N is only a local extremum of Λ(ε),
c m=1.3t 0.03
as it the case, for example, for (U,V) = (0,−2t) and 0.01 m=1.5t
µ = 1.3t as displayed in Fig. 5. Thus, the sign of β xy xy0.02
k k
0.005
0.01
m=1.3t
(a) (U, V )=(-t, 0) (b) (U, V )=(-2t, 0) m=1.5t
1 m=1.3t 1 m=1.3t 00 0.01 0.02 0.03 0.04 00 0.02 0.04 0.06 0.08 0.1
m=1.5t m=1.5t T/t T/t
)e0 e)0 FIG. 7. (Color online) The solid lines, dashed lines and
L( L( dashed-dottedlineswithcirclesstandforκxy,κLxy andκ˜xy as
a function of temperature for (U,V) = (−t,0),(−2t,0) (up-
perpanels)and(U,V)=(0,−t),(0,−2t)(lowerpanels). The
-1 -1
-1 0 1 -1 0 1 black (red) line denotesthe results for µ=1.3t (µ=1.5t).
e /t e /t
(c) (U, V )=(0, -t) (d) (U, V )=(-2t, 0)
3 3
m=1.3t m=1.3t
m=1.5t m=1.5t
2 2 depends on whether |Λ(ε>ε0)| is larger or smaller that
)e e) |Nc|. This behaviorcanbe straightforwardlyreproduced
L( L( using Eqs. (8) and (13). From the above discussion it
1 1
becomesclearthatκ (T,µ)/T woulddisplayastep-like
xy
feature as a function of µ at µ = µ . The deviation
0 0 c
-1 e0 /t 1 -1 e0 /t 1 fromκLxy/T wouldyieldathermalbroadeningofthestep
which would be sharp only in the limit of T =0.
FIG.5. (Color online) Λ(ε)at zero temperaturefor (U,V)= For the purpose of computational feasibility we used
(−t,0),(−2t,0) (upper panels) and (U,V) = (0,−t),(0,−2t) ratherlargeinteractionstrengths,obtaininglargegapsas
(lower panels) at µ=1.3t (black) and µ=1.5t (red). well as short coherence lengths. Our analysis, however,
showsthatthequalitativebehaviorremainsvalid,ifthese
5
parameters are modified towards more realistic values, the thermal Hall effect related to the change of topol-
because the overall features of Λ(ε) remain unchanged. ogy of the superconducting phase. The temperature-
An important point is the change of the Chern number dependenceofthethermalHallconductivityconsistsofa
appearingalsointheplateauofΛ(ε)aroundε=0atthe T-linearwhosecoefficientisuniquelyrelatedtotheChern
Lifshitz transition, µ = µ . Here the gap ε vanishes as number and terms exponentially depending on tempera-
c 0
required for a topological transition. ture. We could show that this latter correction provides
the information on the superconducting gap amplitude
ofthe γ bandasthey areinducedbythermaloccupation
of the quasi-particle states in the bulk continuum. The
IV. CONCLUSIONS
observationofthethermalHalleffectwouldbeapossible
waytofollowthechangesoftopologythroughtheLifshitz
Motivated by the spin-triplet superconductor transitionwhichwouldnotbe possiblebydetecting edge
Sr RuO , we have investigated the interplay be- supercurrents. Our discussion gives a qualitative picture
2 4
tween the thermal Hall effect (Righi-Leduc effect) and what one could observe at a Lifshitz transition which
thetopologyofthechiralp-wavesuperconductingphase. could be possibly induced by doping or uniaxial stress.
We focus on the γ band of Sr RuO which is close to a Clearly a more accurate prediction would be required to
2 4
Lifshitz transition between an electron- and a hole-like include the other two bands which would contribute to
Fermi surface, changingthe topologicalproperties of the thedeviationsfromtheuniversallow-temperaturebehav-
superconducting phase, in particular, its Chern number. ior. It is non-trivial to assess these contributions quan-
The other two Fermi surfaces resulting from the α- titatively, as little is known of the gap structure of these
and β-bands are not affected much by the transition. bands.
Moreover, the Chern numbers of these Fermi surfaces
compensate each other to zero and do not give rise
to a topology related contribution to the thermal Hall ACKNOWLEDGMENTS
coefficient.
WeshowthattheLifshitztransitionintheγ-bandnot We aregratefultoA.Bouhon,J.Goryo,Y.Maeno,T.
only changes the Fermi surface topology but also yields Neupert, T. Saso, A. Schnyder and Y. Yanase for many
a change of the Chern number of the chiral supercon- helpful discussions. The work is supported by the Swiss
ducting state. This alters the structure of the quasi- National Science Foundation and by the Ministry of Ed-
particle edgestates. While the supercurrentsofthe edge ucation,Culture,Sports,ScienceandTechnology,Japan.
states are a surprisingly little affected by this Lifshitz YIisgratefulforhospitalitybythePauliCenterforThe-
transition, our calculations show a strong signature in oretical Studies of ETH Zu¨rich.
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