Disorder-enhanced topological protection and universal quantum criticality in a spin-3/2 topological superconductor

1.1 MB·

by  Sayed Ali Akbar Ghorashi

Disorder-enhanced topological protection and universal quantum criticality in a
spin-3/2 topological superconductor
Sayed Ali Akbar Ghorashi,1 Seth Davis,2 and Matthew S. Foster2,3
1Texas Center for Superconductivity and Department of Physics,
University of Houston, Houston, Texas 77204, USA
2Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
3Rice Center for Quantum Materials, Rice University, Houston, Texas 77005, USA
7 (Dated: April4, 2017)
1
We study the Majorana surface states of higher-spin topological superconductors (TSCs) that
0
could be realized in ultracold atomic systems or doped semimetals with spin-orbit coupling. As a
2
paradigmaticexample,weconsideramodelwithp-wavepairingofspin-3/2fermionsthatgeneralizes

r 3He-B. This model has coexisting linear and cubic dispersing Majorana surface bands. We show
p
thattheseareunstabletointeractions, whichcangenerateaspontaneoussurfacethermalquantum
A
Hall effect (TQHE). By contrast, nonmagnetic quenched disorder induces a surface conformal field
3 theory(CFT)thatisstableagainstweakinteractions: topologicalprotectionisenhanced bydisorder.
Gaplesssurfacestatesofhigher-spinTSCscouldthereforeberobustlyrealizedinsolidstatesystems,

] wheredisorder is inevitable. The surface CFT is characterized byuniversalsignatures thatdepend
l onlyonthebulktopologicalwindingnumber,andincludepower-lawscalingofthedensityofstates,
l
a a universal multifractal spectrum of local density of states fluctuations, and a quantized ratio of
h
the longitudinal thermal conductivity κ divided by temperature T. By contrast, κ /T for the
xx xx
-
cleansurfacewithoutTQHEorderwoulddivergeasT →0. Sincedisorderstabilizestheconducting
s
e Majoranasurfacefluidandquantizesthermaltransport,ourresultssuggestacloseanalogybetween
m bulkTSCs and theinteger quantumHall effect.
.
t
a
m I. INTRODUCTION
-
d The precise quantization of the Hall conductance is u u
n the heart of the integer quantum Hall effect (IQHE).
o The quantization condition holds irrespective of mate- III TQHE
c
rial parameters, quenched disorder, or residual electron-
[
electroninteractions. The robustnessofthe IQHEisdue II SO(n)
2 to the tight entwining of the bulk topology (most easily 4
v Critical
definedforacleansystem),Andersonlocalization(dueto λ
7 quencheddisorder,alwaysrequiredin practice to resolve I Delocalization λ
8
7 the plateaux), and the chiral edge states whose flow can
2 never be degraded by perturbations such as disorder or
0 interactions, but only redirected. Topological protection
. intheIQHElocksaneasilymeasuredobservabledirectly FIG. 1: Schematic renormalization group (RG, left) and pa-
1
rameter (right) phase diagram: 2D Majorana fermion sur-
0 to a bulk winding number.
face fluid of a bulk spin-3/2 topological superconductor with
7 Inthelastdecadetherehasbeenanexplosionofinter-
p-wave pairing and winding number ν = 4. The axes de-
1 estinnewformsoftopologicalmatter,drivenbythedis-
note surface perturbations: the interaction strength u and
:
v coveries of topological insulators and gapless topological quenched disorder strength λ. In the absence of disorder
i phases [1–4]. Despite this progress, a three-dimensional (λ = 0), the clean surface (“I”) is marginally unstable to
X
analog of the IQHE that ties a robust surface transport spontaneoustime-reversalsymmetrybreaking[thermalquan-
r signature directly to a bulk winding number remains tum Hall effect (TQHE) order, (“III”)] for u > 0. Disorder
a
lacking. One potentially promising route is to look for λ > 0 is a strong perturbation that drives the surface into a
a generalization of Helium 3B (3He-B), the only known critically delocalized, time-reversal symmetric state [SO(n)4
CFT, “II”]. The latter is stable against weak interaction ef-
bulk topological superfluid predicted to host a gapless
fects [8]. Disorder is formally irrelevant to III because it can
surface fluid of unpaired Majorana fermions [5, 6].
be viewed as a gapped, paired Majorana BCS condensate.
Previous theoretical work [6–10] has shown that the
RG results near I and II are obtained by analytical and nu-
Majoranasurfacefluidofamodelspin-1/2bulktopolog- mericalcalculations, whileIIIisconfirmedbymean-fieldthe-
ical superconductor (TSC) can be robust to both disor- ory. The boundary between II and III on the right likely
der and interaction effects, and should exhibit a univer- indicates a first order transition, but neither its nature nor
salsurfacethermalconductivityproportionaltothebulk its precise shape in the λ-u plane is determined here. The
winding number [10]. In all of these works, the form of surface thermal conductivity is precisely quantized in both
the 2D surface theory was always assumed to be rela- phases II and III: {κxx,κxy} = {4/π,0}κ◦ in II [6, 10] and
tivistic, with ν “colors”oflinearly-dispersingMajorana {κxx,κxy}={0,±2}κ◦ in III [15–20]. Here κ◦ ≡π2kB2T/6h.
| |
2
fermionscoupledviainteractionsand/orquencheddisor- bulk and the CFT describing the disordered surface of
der; ν denotes the winding number. A key unanswered a TSC, despite the fact that the clean surface theories
question is whether the physics (e.g., universal thermal can fundamentally differ. Technically, it means that the
conductivity) is tied to this simplifying assumption, or topology precisely tunes the surface field theory to the
instead represents a robust aspect of generic bulk TSCs. conformal fixed point made possible by a Wess-Zumino-
Recent theoretical work has turned to higher-spin Novikov-Witten term [8]. Without this fine-tuning, this
TSCs, with potential applications to alkaline and fixed point is unstable to the thermal metal phase (de-
alkaline-earth ultracold atoms [11] or doped semimetals spite the WZNW term) [10].
with spin-orbit coupling [12–14]. In this paper, we con- The SO(n) CFT that describes the disordered Majo-
4
sider the surface states of a spin-3/2 generalization of rana surface fluid is known to be protected againstweak
3He-B with isotropic p-wave pairing [11]. A novel fea- interaction effects [8]. We conclude that disorder stabi-
ture is that the surface Majorana fluid exhibits coexist- lizes the surface Majorana fluid for this spin-3/2 model,
inglinearandcubicbands. Cubicsurfacebandswerealso andthisimpliesthathigher-spinTSCscouldberobustly
predictedinacloselyrelatedmodel[12]thatmayberele- protected. ThethermalHallconductivityκ dividedby
xy
vant for superconducting half-Heusler alloys. Due to the temperatureT ofthesurfaceTQHEisquantizedanduni-
vanHovesingularity,one mightexpectthat anyresidual versal [15–20]: κ = Wκ , where W Z is the surface
xy ◦
∈
interactions between surface Majorana particles would winding number and
produce a strong instability. Surprisingly, we show that
interactionsareonlymarginally relevant: onlyattractive κ /T =π2k2/6h. (1.1)
◦ B
interactions induce spontaneous time-reversal symmetry
breaking and lead to a surface thermal quantum Hall ef- What is more important here is that [6, 10]
fect (TQHE) [6, 7]. This weak instability is tied to the
κ ν κ
strong constraints imparted by Pauli exclusion to a Ma- xx ◦
lim = | | , κ =0, (1.2)
xy
jorana gas, despite the density of states divergence. Re- T→0 T π T
pulsive interactions are marginally irrelevant;their main
for the disorder-induced surface CFT (which preserves
effect would be to generate a finite longitudinal surface
time-reversal symmetry). Here the winding number ν =
thermal conductivity κ at temperature T > 0 due to
xx 4. Eq. (1.2) implies that the low-temperature thermal
inelasticscattering. Intheabsenceofimpurityscattering
conductivity is quantized by the bulk winding number,
the ratio κ /T would diverge as T 0.
xx → independentofbothdisorderandinteractions. Sincedis-
Bycontrast,nonmagneticquencheddisorderprovesto
orderstabilizesthesurfaceandinducesaquantizedther-
be a strong perturbation. Using exact diagonalization
mal conductivity, bulk TSCs appear to be closely anal-
to study the noninteracting dirty surface, we show that
ogous to the integer quantum Hall effect in two dimen-
disorderinducesscalingconsistentwithacritical,exactly
sions. Our results are summarizedby the phase diagram
solvable conformal field theory (CFT) SO(n) [8]. (Here
4 in Fig. 1.
n 0denotesthenumberofreplicas.) TheCFTgoverns
→ This paper is organized as follows. In Sec. II, we de-
the divergence of the global density of states and the
fine the bulk model and describe the form of the sur-
statistics of the single-particle wave functions. The level
face states. We then summarize our results regarding
of the current algebra (=4) is also the modulus of the
themarginalinstabilityofthecleansurface,andtheuni-
bulk winding number ν for our p-wave model. This is
| | versalquantumcriticalityofthedisorderedone. Therest
the same result that obtains for spin-1/2 models of TSC
ofthepaperexplainskeytechnicaldetails. Sec.IIIshows
surface states studied previously [8, 10]. In the spin-1/2
the derivation of the surface state Hamiltonian and the
case with winding number ν, the clean surface fluid is
calculationofthesurfacewindingnumberinthepresence
a free fermion (level one) CFT due to the relativistic
of explicit time-reversal symmetry breaking. The effects
dispersion. TheemergenceofanotherCFTwithlevel ν
| | ofinteractionsonthecleansurfacearetreatedusingone-
in the presence of disorder follows from certain rules in
loop renormalization and mean field theory. Sec. IV de-
these theories (conformal embeddings [8, 10]).
scribestheincorporationofdisorder,andprovidesdetails
Here the situation is very different. The clean Majo-
of the numerical diagonalization scheme.
rana fluid of the spin-3/2 model is not a CFT, as evi-
denced by the cubic dispersion. Moreover, a standard
derivation of the effective surface theory with disorder
wouldincorrectlypredict a thermal metalwith weak an-
tilocalization [8, 21]. Properties of this metal would de- II. MODEL AND RESULTS
pendonthebaredisorderstrengthandwouldvaryslowly
with system size. Our numerics instead show univer-
A. Bulk and surface models
sal scaling that is independent of the disorder strength.
In conjunction with the conformal embedding argument
for the spin-1/2 case, the results obtained here empiri- We consider a system of spin-3/2 fermions. In the ab-
callysuggesta deeprelationbetweenthe topologyofthe sence of pairing, we assume a bulk Hamiltonian of the
3
form agation parallel to the surface. For k =0 and hard wall
boundary conditions, there are four zero energy bound
d3k 2
H0 = (2π)3 c†(k) k2−γ Sˆ·k c(k), (2.1) sptaartteicsl{e-|ψho0,lmes“is}p.inT”hleocmkossttoimthpeoprthaynsticfaelastpuirneaisttthhaetstuhre-
Z (cid:20) (cid:16) (cid:17) (cid:21)
face:
where k = k . The 4-component fermion field
x,y,z
{ }
c(k) c (k) has Sˆz-label m 3,1, 1, 3 ; Sˆ = ψ = σ1 =sgn(m ) m f . (2.6)
→ ms s ∈ {2 2 −2 −2} | 0,msi | s i⊗| si⊗| msi
Sˆx,y,z are spin-3/2 operators. Eq. (2.1) is an isotropic
v{ersion}of the Luttinger Hamiltonian [22] used to model The particle-holespinpoints alongthe +σ1 ( σ1)direc-
−
heavy and light hole bands in zinc-blende semiconduc- tion for positive (negative) ms. In Eq. (2.6), hz|fmsi =
tors; the parameter γ measures the strength of effective fms(z) denotes the bound state envelope function.
spin-orbit coupling (SOC) amongst the states of the 3/2 Since time-reversal gives the effective chiral condition
multiplet. Here we have set 2m=1 in the first term (m σˆ1hˆ(k)σˆ1 = hˆ(k) in the bulk, the locking condition
−
isthebandmassintheabsenceofSOC).Weassumethat impliesthatthe effectivesurfaceHamiltonianhˆ satisfies
S
γ < 4/9, so that both bands “bend up.” The situation
where bands bend oppositely is relevant for half-Heusler τˆ3hˆ (k)τˆ3 =hˆ (k), (2.7)
S S
−
alloys; in that case similar Majorana surface states can
arisewithbulkp-wavepairing[12],butthewindingnum- where τˆ3 = +1 ( 1) for m > 0 (m < 0). We in-
s s
−
bers differ [23]. troduce two mutually commuting species of Paulimatri-
ces: τˆ1,2 anticommute with τˆ3 andactonthe sgn(m )
We assume isotropic p-wave pairing of spin-3/2 s
space{, wh}ile κˆ1,2,3 mix the 3/2 and 1/2 states with
fermions [11]:
{ }
the same sign. E.g., τˆ3 = diag(1,1, 1, 1), κˆ3 =
− −
1 d3k diag(1, 1,1, 1).ThematrixRˆ =iτˆ1κˆ2. Thenthelock-
H = χ†(k)hˆ(k)χ(k), (2.2) − −
2 (2π)3 ing condition implies the automatic surface particle-hole
Z symmetry
where the 8 8 Bogoliubov-de Gennes Hamiltonian is
× Mˆ(S)hˆT( k)Mˆ(S) =hˆ (k), Mˆ(S) =τˆ2κˆ2. (2.8)
2 − P S − P S P
hˆ(k)= k2 µ γ Sˆ k σˆ3+∆ Sˆ k σˆ2. (2.3)
− − · p · The form of hˆ (k) is constrained by Eqs. (2.7) and
(cid:20) (cid:16) (cid:17) (cid:21) (cid:16) (cid:17) S
(2.8), as well as rotational invariance in the plane. An
Here µ is the chemical potential and ∆ the BCS gap
p explicit kp calculation gives the low-energy form [11]
parameter. The 8-component field in Eq. (2.2) has the ·
particle-hole space decomposition hˆ (k)= i τˆ+κˆ−k τˆ−κˆ+¯k + c τˆ+k2+τˆ−¯k2
S 4 − 2
c(k) (cid:0)0 0 ck2 0(cid:1) (cid:0) (cid:1)
χ(k)≡(cid:20)(−iRˆ) c†(−k) T(cid:21), (2.4) = c0¯k2 0i¯k i0k c0k2. (2.9)
where T denotes the transpos(cid:2)e in spi(cid:3)n-3/2 space. The  0 −c¯k2 0 0 
 
Pauli matrices σˆ1,2,3 in Eq. (2.3) act on particle-hole  
space. In Eq. (2.4), Rˆ is an antisymmetric 4 4 matrix Here, k,¯k = kx iky, τˆ± = τˆ1 iτˆ2, κˆ± = κˆ1 iκˆ2,
{ } ∓ ± ±
× the coefficient for the linear dispersion is normalized to
satisfying
one, and c is a real constant with units of length. For
Rˆ(Sˆ)TRˆ−1 = Sˆ, Rˆ2 = ˆ1. (2.5) weak SOC (0< γ 1), it is easy to see that c γ/∆p.
− − Without SOC, sec≪ond-order kp theory gives c∝= 0 so
·
that the m = 3/2 bands remain flat. Nonzero c is
The field χ satisfies the “Majorana” condition s ±
χ†(k) = iχT( k)MˆP, where MˆP = iσˆ2Rˆ = symmetry-allowedandthus expected in the genericsitu-
MˆT. This impli−es the automatic particle-ho−le symme- ation; an alternative route incorporates additional small
P s-wave pairing. In Eq. (2.9), we neglect terms cubic in
try MˆP−1ˆhT( k)MˆP = hˆ(k). Time-reversal invariance k because these do not modify the low-energy disper-
is en−coded as−MˆT−1hˆT( k)MˆT = hˆ(k), where MˆT = sixo,ny relations.
−
MˆTT =σˆ3Rˆ. Combiningparticle-holeandtime-reversal In the limit k 0, Eq. (2.9) exhibits linear and cubic
[−6, 11] gives the effective chiral condition σˆ1hˆ(k)σˆ1 = bands [11]: →
−
hˆ(k). With all of these symmetries the model belongs to
k
class DIII [6]. The bulk winding number is ν =4 [11] so ε (k)= 1+4c2k2+1 k+O(k)3,
1
2 ≃
long as µ>0 and γ <4/9. (2.10)
k (cid:16)p (cid:17)
The effective surface Hamiltonian obtains by termi- ε3(k)= 1+4c2k2 1 c2k3+O(k)5.
2 − ≃
nating the system in the z-direction and diagonalizing
(cid:16)p (cid:17)
hˆ(k ,k , i∂ ). The momentum k= k labels prop- Both bands become quadratic at large k. Due to the
x y z x,y
− { }
4
cubic band, the systemhasa vanHovesingularityinthe Majorana fluid carefully. This is because the latter is
density of states ν(ε 0) ε−1/3. gapless and exhibits a diverging density of states. In a
→ ∼
superconductor, interactions at the surface are expected
to be short-ranged due to screening by the bulk. These
B. Surface perturbations residual interactions can be mediated by virtual fluctu-
ations of the “massive” electromagnetic field. Here we
The Hamiltonian for the surface fermion fluid is given posit the form of the interactions based on symmetry
by and Pauli exclusion.
Because η is a four-component Majorana field, there
H(S) = 1 d2rηTMˆ(S)hˆ η, (2.11) is only a single interactionterm without derivatives that
0 2 P S we can write; others with derivatives are less relevant.
Z
Labeling the components as η η ,
1,2,3,4
where η η is a four-component Majorana spinor →
and r is t→he pomssition vector. Local bilinear (“potential”)
H(S) u d2rη η η η
perturbationsmustobey surfaceparticle-holesymmetry. I ≡ 1 2 3 4
There are 6 Hermitian terms without derivatives of the Z
u
form 12 d2rηTMˆP(S)Λˆη, where Λ ∈ {τˆ1,2,3} or {κˆ1,2,3} = ∓ 8 d2r(ηTMˆP(S)Λˆη)2, (2.12)
satisfies Eq. (2.8). These are classified by symmetry. Z
UndeRr planar rotations, τˆ3 and κˆ3 are scalars, κˆ1,2 where the minus (plus) sign corresponds to Λˆ =τˆ3 (κˆ3).
transform like a vector, and τˆ1,2 transform like second-
Thus u>0 is anattractive (repulsive) interactionin the
ranktensorcomponents. Onlyτˆ1,2aretime-reversaleven
τˆ3 (κˆ3)channel. Thecouplinguhasunitsoflength. The
[Eq. (2.7)]; the rest are odd.
signofucouldbedeterminedbyintegratingoutthebulk
AgenericcombinationΛˆ =m1τˆ3+m2κˆ3 breakstime- superfluid and the electromagnetic field, but we will not
reversal and induces a gapped surface thermal quantum do so here.
Hall (TQH) state [3, 16]. We compute the surface wind-
Given that the noninteracting surface fluid has coex-
ing number W using the Green’s function [5]; the result
isting linear and cubic bands, it is not a priori obvious
is shown in Fig. 2. The lines m = m are gapless sur-
1 2 how to assess the relevance of u from a renormalization
±
face plateau transitions. The maximum winding num-
group (RG) perspective. Moreover, the van Hove singu-
ber and surface gap for fixed m2+m2 is achieved for
1 2 larity suggests that nonzero u will induce bad infrared
m > m , i.e. τˆ3 order. Finally, we note that the spin
| 1| | 2| p behavior. In fact one-loop perturbation theory gives the
operator Sˆz corresponds to m1 =2m2, so that an exter- simple vertex correction,
nal Zeeman field would induce the W = 2 plateaux.
±
Γ(4) = u+ u2/4πc ln(4cΛ) , (2.13)
−
C. Marginal instability of the clean surface where Λ is the ultra(cid:2)violet(cid:0)momen(cid:1)tum cut(cid:3)off. The cor-
rection is only logarithmic, and is cut in the infrared by
the length scale c. This immediately implies the beta
Although we treat the gapped bulk as an effectively
function,
non-interacting mean field Hamiltonian, we must con-
sider the effects of residual interactions on the surface
du˜ u˜2
= +O u˜3 , (2.14)
dl 4π
m
(cid:0) (cid:1)
2 where u˜=u/c is the dimensionless coupling.
1 TheabsenceofbadinfraredbehaviorinEq.(2.13)and
theweaknessoftheultravioletsingularityisduetoPauli
m exclusion, i.e. the fact that both linear and cubic com-
-2 2
1 ponents of the Majorana Green’s function must appear
simultaneouslyintheloop. Eq.(2.14)impliesthatu>0
-1
isamarginallyrelevantperturbation. Eq.(2.12)suggests
anaturalinterpretationinterms ofTQHorderwithsur-
face winding number W = 2.
±
FIG.2: Phasediagramofthecleanspin-3/2Majoranasurface We can confirm this picture with a mean-field calcu-
fluidinthepresenceoftime-reversalsymmetry-breakingmass lation. We decouple the interaction in Eq. (2.12) with
termsm1 andm2. Allstates areTQHEplateaux with wind- the order parameter (u/2) ηTMˆ(S)τˆ3η . Zero-
ing number as indicated, computed via the surface Green’s M ≡ h P i
temperature mean-field theory gives
function [5]. The combination m2 = m1/2 corresponds to
the spin-3/2 operator Sˆz, as could be introduced via Zee-
3
1 4.4 1
man coupling to an external magnetic field. The conditions , u˜ 1. (2.15)
m1 =±m2 are gapless plateau transitions. M≃ c 2π [(2π/u˜) ln(cΛ)]3 ≪
(cid:18) (cid:19) −
5
Physicallywecanassociatenonzero withsurface “i though we are confident that the surface resides in the
M
s” (imaginary s-wave) pairing of the Majorana particles W = +2 or 2 plateau for u > 0, there are hints that
−
[7, 8, 24]. This can be understood via the following ar- mean-fieldtheoryfailstocorrectlypredicttheadmixture
gument. In the bulk, one can write a local spin singlet, of m and m . For details, see Sec. IIID, below.
1 2
time-reversalodd pairing operator
T
[ ic†Rˆ c† +H.c.]=χ†σˆ1χ. (2.16)
− D. Quenched disorder and universal surface
Eq. (2.6) implies (cid:0)th(cid:1)at this bilinear projects to quantum criticality
ηTMˆ(S)τˆ3η at the surface. On the other
P
∝ M
hand, Eq. (2.12) can also be written as proportional In a solid state realization, quenched disorder due to
to u d2r(Sˆz)2, implying surface magnetic order for impurities and other defects is inevitable at the sample
u >− 0. Indeed, these disparate orders are unified in surface. Now we consider the effects of disorder on the
the surRface fluid, due to the strong spin-orbit coupling non-interacting surface states.
in the bulk and the locking condition [Eq. (2.6)]. We We add real disorder potentials that couple to the
might anticipate a generic order parameter of the form time-reversal symmetric bilinear perturbations τˆ1,2 to
Λˆ = m τˆ3 +m κˆ3 [c.f. Fig. 2] with m > m . Al- Eq. (2.9):
1 2 1 2
| | | |
0 0 c( i∂)2+P(r) 0
0 0 − ∂ c( i∂)2+P(r)
hˆS →hˆS+P1(r)τˆ1+P2(r)τˆ2 =c( i∂¯)2+P¯(r) ∂¯ 0 − 0 , (2.17)
 − 0 c( i∂¯)−2+P¯(r) 0 0 
 − 
 
where ∂,∂¯ =∂ i∂ and P,P¯ =P iP . We con- findgoodagreementirrespectiveofthedisorderstrength.
x y 1 2
{ } ∓ { } ∓
sider only time-reversal invariant disorder; equivalently, The disorder-induced spatial fluctuations of the criti-
we require that there are no magnetic fields or magnetic
impurities at the surface. We assume Gaussian white
noise disorder potentials P1,2(r) with common variance N(ε)/N(ε∗)
given by a dimensionless parameter λ. Due to the cu- λ = π
bic dispersion, even weak disorder is expected to pro-
duce a strong effect. Indeed, perturbative renormaliza- λ = 4π/5
tion of λ produces a quadratic infrared divergence, and
implies that λ/c2 is the effective disorder strength. This λ = 3π/5
has dimension 2 and thus corresponds to a strongly rel-
λ = 2π/5
evant perturbation of the clean surface band structure.
To treat the disorder nonperturbatively, we diagonalize
λ = π/5
Eq. (2.17) numerically. The calculation is performed in
momentum space to avoid fermion doubling issues [9]. ε/ε∗
Details are described in Sec. IVB, below.
In spin-1/2 bulk TSCs with relativistic surface fluids,
conformal embedding rules establish certain 2+0-D con- FIG.3: Numerical evidenceforcritical surfacedelocalization
formal field theories (CFTs) as governing the properties in the presence of quenched disorder I: Integrated local den-
of disordered, noninteracting surface states [8]. For a sityofstates(IDoS)N(ε). TheexactpredictionoftheSO(n)4
classDIIIbulkwithwindingnumber4,thattheorywould theory gives N(ε)∼ε4/5 (blue solid lines). The clean theory
predict the surface CFT SO(n) , where n 0 counts has N(ε) ∼ ε2/3 due to the van Hove singularity. Data (red
4
→ dotted lines) is obtained from momentum-space exact diag-
replicas. We will demonstrate that this theory also gov-
onalization of the dirty surface Hamiltonian, without inter-
erns the dirty surface states of the spin-3/2 TSC.
actions. Results are presented for typical realizations of the
The SO(n)4 theory predicts [8] a diverging low-energy disorder(i.e.,thereisnodisorder-averaging). Curveswithdif-
global density of states (DoS) ν(ε) ε−1/5. Note that ferent disorder strengths λ are labeled and shifted vertically
∼
this is a weaker power law than the 1/3 van Hove singu- for clarity. The system size consists of an 81 × 81 grid of
larityinthecleansystem. InFig.3,wecomparenumeri- momenta. Irrespective of the nonzero disorder strength, the
calresultsfordifferentdisorderstrengthstotheCFTpre- same critical scaling exponent is observed and is consistent
diction for the integrated DoS N(ε) εdε′ν(ε′). We with theSO(n)4 theory.
≡ 0
R
6
τ(q) theinteractionstrength,inafixedrealization ofdisorder,
(i) and (2) within a disorder-averaged large winding num-
(ii) ber expansion. These give the same result only if the
(iii) q disorder-averagedsystem is tuned to the CFT (in which
(iv)
case Altshuler-Aronov corrections vanish) [10]. We do
(v)
nothavethe conformalembedding dictionary[8]utilized
(i) λ = π for spin-1/2 TSCs, but our numerical results empirically
suggest that there is an equivalence between the bulk
(ii) λ = 4π/5
topology and the CFT describing the disordered surface
(iii) λ = 3π/5
of a TSC [8–10], despite the fact that the clean surface
(iv) λ = 2π/5 theories can fundamentally differ.
(v) λ = π/5
FIG. 4: Numerical evidence for critical surface delocaliza-
tion in the presence of quenched disorder II: Multifractal
spectrum. The exact prediction of the SO(n)4 theory gives E. Stability, phase diagram, and quantized thermal
Eq. (2.18) (blue). The clean theory would have τ(q) = conductivity
2(q−1). Data [red, labeled (i)–(v)] is obtained as in Fig. 3.
The disorder strengths λ are indicated for the numerical
curves. Curves with different disorder strengths are shifted We have seen that the clean surface is marginally un-
verticallyforclarity. Thelargestdeviationoccursfor|q|>q , stabletoTQHorder. We havealsoshownthatquenched
c
whereq =2isthemultifractalterminationthreshold[25–27]. disorder is a strong perturbation that drives the non-
c
Thisisafiniteresolutioneffect,sincetheslopesbeyondqc are interacting surface to a phase described by the SO(n)4
governed by thepeaks and valleys of thewave function. CFT. It is known [8] that interactions are strongly irrel-
evant to this CFT,
du˜ u˜
calsurfacewavefunctionsareencodedinthemultifractal = +O u˜2 , (2.19)
spectrumτ(q). (Forarecentreviewonmultifractalityat dl −2
Anderson metal-insulator transitions, see e.g. Ref. [25].) (cid:0) (cid:1)
where u˜ is the dimensionless coupling strength. Al-
The SO(n) theory predicts an exactly quadratic spec-
4 though multifractality can sometimes enhance interac-
trum [8] for the low-energy wavefunctions,
tions [7, 28, 29], that does not occur here. The reasonis
again Pauli exclusion: the interaction and second multi-
τ(q)=(q 1)(2 q/2), q 2. (2.18)
− − | |≤ fractalmomentoperatorsaredistinctduetothecomplete
antisymmetrization of the former [8]. We therefore con-
Fig. 4 compares Eq. (2.18) to the numerical results.
clude that disorder stabilizes the surface Majorana fluid
Figs. 3 and 4 provide strong evidence that the dis-
of this spin-3/2 TSC. This is our most important result.
ordered, noninteracting spin-3/2 Majorana fluid is gov-
erned by the SO(n) theory. This is surprising because We note that Eq. (2.19) technically obtains from the
4
a standard derivation [21] of the disorder-induced effec- dynamical version of the SO(n)4 theory. This is a 2+1-
tive field theory would predict a thermal metal phase DtheoryofMajoranafermionspropagatinginspaceand
exhibiting weak antilocalization. Although the theory time, whose disorder-averaged spatial correlations and
in[21]shouldbeaugmentedbyaWess-Zumino-Novikov- dynamical scaling exponent are governed by the 2+0-
Wittenterm(seeSec.IVAfordetails),thistermdoesnot D replicated CFT, see [8] for details. Thus in Fig. 1,
alter the tendency towardsantilocalizationin the metal- “SO(n)4” really refers to this dynamical hybrid theory,
licphase[8]. Becausethecleandensityofstatesdiverges which can also be expressed as a Wess-Zumino-Novikov-
for the surface Majorana fluid studied here, a diffusive Witten Finkel’stein nonlinear sigma model (WZNW-
metallic state would be generically expected. Yet this FNLsM) [8, 10].
is inconsistent with our numerical results [which instead The thermal conductivity of the WZNW-FNLsM re-
matchtheSO(n)4 CFT].Animportanttechnicalpointis ceives no quantum interference corrections due to dis-
thattheCFTisunstable tothethermalmetalphase[see order [6, 10] at the conformal fixed point. Interaction-
Eq.(4.1)]. Itmeansthatthe CFT canonlybe realizedif mediated Altshuler-Aronov corrections also vanish to at
the system is tuned to the SO(n)4 fixed point. Our nu- least order 1/ν , where ν is the bulk winding number.
merical results imply that this is exactly what happens. The absence o|f F|riedel oscillations in any Majorana sur-
The same “fine-tuning” is required for the spin-1/2 face “density” implies that these should be absent to all
TSCs. In that case, however, there is a nonperturba- orders [10]. Since the spin-3/2 Majorana surface fluid
tive argument for it using conformal embedding theory studied here realizes the SO(n) theory in the presence
4
[8]. Additional evidence in the spin-1/2 case obtains by of disorder, we conclude that the ratio of the longitu-
comparing interaction (Altshuler-Aronov) corrections to dinal thermal conductivity and temperature is precisely
transport via two methodologies: (1) order-by-order in quantized as T 0 [Eq. (1.2)].
→
7
F. Summary the surface and induces a thermal quantum Hall effect
(TQHE) plateau. By contrast, quenched disorder gives
In summary, we have derived surface states and sur- the SO(n)4 theory previously predicted for a spin-1/2
face effective Hamiltonian for a spin-3/2 time-reversal TSC with winding number 4; this theory is stable to in-
invariant topological superconductor that hosts a cubic teraction effects. We conclude that disorder enhances
dispersion coexisting with the conventional linear Majo- topologicalprotection. Inthe low-temperaturelimit, the
ranacone. We haveshownthatin the cleanlimit, unlike ratioofthelongitudinalthermalconductivitytotemper-
the spin-1/2 case [30], interactions are marginally rele- ature is predicted to be quantized and proportional to
vant and lead to a BCS-type instability that gaps out the bulk winding number, as shown in Eq. (1.2).
III. IDEAL (CLEAN) SURFACE STATES AND INTERACTIONS
A. Derivation of the surface Hamiltonian Eq. (2.9)
1. Luttinger Hamiltonian bulk
To obtain the surface Hamiltonian, we consider a bulk superconductor in the half space z 0 with hard wall
≥
boundary conditions. We divide Eq. (2.3) into two parts:
ˆh=hˆ ( i∂ )+hˆ (k, i∂ ),
0 z 1 z
− −
hˆ =σˆ3 ∂2 µ +σˆ2 ∆ Sˆz( i∂ )+∆ , (3.1a)
0 − z − p − z s
(cid:0) (cid:1)1 h i 2 ∆
hˆ =σˆ3 k2 γ Sˆ+k+Sˆ−¯k +Sˆz( i∂ ) +σˆ2 p Sˆ+k+Sˆ−¯k , (3.1b)
1 z
( − (cid:20)2(cid:16) (cid:17) − (cid:21) ) 2 (cid:16) (cid:17)
where k= k is the momentum parallelto the surface, k,¯k =k ik , and Sˆ± =Sˆx iSˆy are spin-3/2 raising
x,y x y
{ } { } ∓ ±
and lowering operators. In Eq. (3.1a), we have added a time-reversal symmetric s-wave pairing term proportional to
∆ . We will utilize this below in the case of vanishing bulk spin-orbit coupling (SOC). Energies like ∆ and µ have
s s
units of 1/(length)2, while ∆ has units of 1/length.
p
In this subsection we will ignore s-wave pairing (∆ = 0) and we will treat the SOC term proportional to γ as a
s
small perturbation (although this is not necessary). The surface eigenstates of hˆ with zero transverse momentum
0
satisfy hˆ ψ =0 and take the form shown in Eq. (2.6),
0| 0,msi
1 1 ∆p ms z ∆2pm2s
ψ (z)=f (z) m , f (z)= exp | | sin z µ . (3.2)
0,ms ms (cid:20)sgn(ms)(cid:21)| si ms Nms (cid:18)− 2 (cid:19) " r − 4 #
p
Here the explicit 2-component spinor resides in particle-hole (σ) space; the four zero energy states are distinguished
by their Sˆz eigenvalues m 3, 1 . The particle-hole spinor is “locked” to the physical spin, as it points along
s ∈ {±2 ±2}
the +σ1 ( σ1) direction for positive (negative) m . The form of the envelope function f (z) is appropriate for the
− s ms
weak pairing limit, (3∆ /4)2 <µ.
p
To obtain the effective surface Hamiltonian for nonzero transverse momentum, we diagonalize hˆ in the basis of
1
zero modes given by Eq. (3.2). The only non-vanishing elements obtain from
∆ γ
hˆ σˆ2 p Sˆ+k+Sˆ−¯k σˆ3 (Sˆ+)2k2+(Sˆ−)2¯k2 . (3.3)
1
→ 2 − 4
(cid:16) (cid:17) h i
The first term connects the 1/2 states, giving the k,¯k -linear terms in Eq. (2.9). The second term mixes the
3/2, 1/2 and 1/2, 3/2 ±states, giving the k2,¯k2{ter}ms in Eq. (2.9). The parameter c γ/∆ .
p
{ − } { − } { } ∝−
2. Vanishing SOC in the bulk
If the SOC parameter γ =0, then the 3/2 surface bands remain flat in degenerate perturbation theory. Another
±
waytogetnonzerocinEq.(2.9)isbyincorporatinganadditionalweaks-wavepairingamplitude∆ ,asinEq.(3.1a).
s
8
ThenthezeroenergysurfaceeigenstateswithvanishingtransversemomentumagaintaketheformshowninEq.(2.6),
but with the modified envelope function
1 ∆ m z ∆2m2
f (z)= exp p| s| sinh z p s µ i∆ sgn(m ) . (3.4)
ms Nms (cid:18)− 2 (cid:19) " r 4 − − s s #
p
Here we assume that 0<∆ ∆2 (so as to remain in the bulk topological phase with winding number ν =4 [11]).
s ≪ p
For convenience, we also assume intermediate strength pairing such that 0 < µ < (∆ /4)2; in this case there is only
p
one branch of bulk scattering states.
To obtain the effective surface Hamiltonian for nonzero transverse momentum, we use kp theory. The matrix
elements of hˆ [Eq. (3.1b) with γ =0] give the k,¯k -linear terms in Eq. (2.9), which connect·the m = 1/2 states.
1 s
{ } ±
To connect the 3/2 states to the former, one has to go to second order. This yields the matrix elements [31]
±
−hψ0,ms|hˆ1(k)Pˆhˆ−01Pˆhˆ1(k)|ψ0,m′si, (3.5)
wherePˆ projectsoutofthe degeneratezeromodespace. Eq.(3.5)canbe expressedusingthe basisofbulk scattering
states with zero transverse momentum:
hψ0,ms|hˆ1(k)Pˆˆh−01Pˆhˆ1(k)|ψ0,m′si=Xm′s′ Z0∞ εmd′s′q(q)" h+ψ0h,ψm0s,|mhˆs1|(hˆk1)(|kψ)q,σˆm2′sR′ˆih|ψψq∗q,,mm′s′s′′i|hhˆψ1(q∗k,m)′s|′ψ|σˆ0,2mRˆ′sihˆ1(k)|ψ0,m′si#, (3.6)
where ψq,m′′ denotesascatteringstatewithstandingwavemomentumq(oscillationinthez-direction),Sˆz-eigenvalue
| si
m′s′, and gapped positive energy eigenvalue εm′s′(q), while σˆ2Rˆ|ψq∗,m′s′i is the (negative energy) particle-hole conjugate
of ψq,m′′ . The matrix Rˆ was introduced in Eq. (2.5).
| si
The scattering states take the form
1
ψq,ms(z)= αˆq,ms cos(qz)−e−λq,msz +βˆq,ms sin(qz) |msi, (3.7)
Nq,ms n (cid:2) (cid:3) o
where αˆ and βˆ are 2-compponent spinors in particle-hole space. The expressions for these and λ are
q,ms q,ms q,ms
unwieldy so we omit them here.
Finally, one computes Eq. (3.6) using Eqs. (2.6), (3.4), and (3.7). This second-order result vanishes for ∆ = 0.
s
NonzerocinEq.(2.9)issymmetryallowed,andthusexpectedinthegenericsituation. Thesimplestwaytogetitisby
retainingnonzero∆ intheboundstates ψ ,butneglectingitinthescatteringstates ψ (whichbecome
s {| 0,msi} {| q,msi}
very complicated for ∆ =0). This gives nonzero terms mixing the 3/2, 1/2 and 1/2, 3/2 states proportional
s
6 { − } { − }
tok2 and¯k2 inhˆ (aboveandbelowthediagonal,respectively,consistentwithplanarrotationalinvariance). Without
S
loss of generality, we can take the coefficients to be real and positive since the phases can be removedwith a unitary
transformation.
B. Calculation of the surface winding number in Fig. 2
We compute the surface winding number W for the clean, noninteracting Majorana surface fluid perturbed by
time-reversalbreaking “mass” terms. The Hamiltonian is
hˆ (k) hˆ (k)+m τˆ3+m κˆ3, (3.8)
m1,m2 ≡ S 1 2
where hˆ was defined by Eq. (2.9). The energy bands of hˆ (k) are gapped for non-zero values of m unless
m = mS , in which case a gapless linear Dirac point appeamrs1,amt2k=0. 1,2
1 2
±
Since the mass terms break surface time-reversalsymmetry [Eq. (2.7)], the surface theory resides in class D [6]. In
2D, this class can exhibit a thermal quantum Hall effect [15–17], where edge states carry a quantized energy current.
The thermal Hall conductivity κ can be expressed in terms of a winding number W via [3, 18]
xy
κ =Wκ , (3.9)
xy ◦
9
FIG. 5: Feynmandiagrams for theone-loop vertexcorrections.
where κ was defined by Eq. (1.1). In terms of the surface Green’s function
◦
−1
Gˆ(ω,k,m ,m ) iωˆ1+hˆ (k) , (3.10)
1 2 ≡ − m1,m2
h i
the winding number is given by [5]
ǫ ∞
W(m ,m ) αβγ dω d2k Tr Gˆ−1∂ Gˆ Gˆ−1∂ Gˆ Gˆ−1∂ Gˆ , (3.11)
1 2 ≡ 3!(2π)2 Z−∞ ZR2 h(cid:16) α (cid:17)(cid:16) β (cid:17)(cid:16) γ (cid:17)i
where Tr denotes the trace over spin-3/2 components and α,β,γ ω,k ,k . Numerical evaluation of Eq. (3.11)
x y
∈ { }
using Eqs. (3.8) and (3.10) leads to the winding number results shown in Fig. 2.
C. Perturbative vertex renormalization
The imaginary time action for the clean, time-reversal invariant, interacting Majorana surface theory implied by
Eqs. (2.9) and (2.12) is given by
1 dωd2k u
S = ηT( ω, k)Mˆ(S) iω+hˆ (k) η(ω,k)+ dτd2rǫ η η η η , (3.12)
2 (2π)3 − − P − S 4! i1i2i3i4 i1 i2 i3 i4
Z h i Z
where we have antisymmetrized the four-fermion interaction using the fourth-rank Levi-Civita tensor. Repeated
indices are summed.
To one loop, the bare vertex function evaluates to
u2
(Γ(4)) = uǫ + [ǫ ǫ +ǫ ǫ +ǫ ǫ ]
i1i2i3i4 − i1i2i3i4 2 i1i2j1j2 j3j4i3i4 i1i3j1j2 j3j4i4i2 i1i4j1j2 j3j4i2i3
dωd2k
GˆMˆ(S) (ω,k) GˆMˆ(S) (ω,k), (3.13)
×Z (2π)3 h P ij1j3 h P ij4j2
valid in the limit of vanishing external frequencies and momenta. The three double Levi-Civita terms in the square
bracketscorrespondto the three loopcorrectionsshowninFig.5. We emphasize thatthe signofeachdiagramhasto
be carefully determined using Wick’s theorem for the Majorana fermion field. The Green’s function Gˆ(ω,k) is given
by Eq. (3.10) with m =m =0, while Mˆ(S) was defined by Eq. (2.8).
1 2 P
We define
D(ω,k) c4k8+2c2k4ω2+k2ω2+ω4
≡
and
2π
(ω,k) dφ D2(ω,k) GˆMˆ(S) (ω,k,φ ) GˆMˆ(S) (ω,k,φ ), (3.14)
Ni1i2i3i4 ≡Z0 k h P ii1i2 k h P ii3i4 k
where we have switched to polar momentum coordinates k (k,φ ). Next we compute
k
→
1
[ǫ ǫ +ǫ ǫ +ǫ ǫ ] (ω ck2x,k)
2 i1i2j1j2 j3j4i3i4 i1i3j1j2 j3j4i4i2 i1i4j1j2 j3j4i2i3 Nj1j3j4j2 ≡
= 4c4k10π x2+c2k2(1+x2)2 ǫ . (3.15)
− i1i2i3i4
(cid:2) (cid:3)
10
Thus Eq. (3.13) reduces to
∞ Λ (ck3)4c4k10π x2+c2k2(1+x2)2
(Γ(4)) = ǫ u +u2 dx dk
i1i2i3i4 − i1i2i3i4( Z−∞ Z0 23π3 (cid:2) D2(ck2x,k) (cid:3))
u2
= ǫ u + ln (2cΛ)2+1+2cΛ , (3.16)
− i1i2i3i4 4πc
(cid:26) hp i(cid:27)
where Λ denotes the ultraviolet momentum cutoff. Taking the limit cΛ 1 gives Eq. (2.13).
≫
D. Mean-field theory: surface thermal quantum Hall plateaux
The interaction strength u is enhanced (suppressed) by quantum fluctuations for u > 0 (u < 0) [Eq. (2.14)]. In
Eq. (2.12) and the text following, it is noted that u > 0 is an attractive (repulsive) interaction in the “τˆ3” (“κˆ3”)
channel, where these matrices specify mass terms used to construct the thermal quantum Hall phase diagram shown
in Fig. 2. We therefore expect that spontaneous symmetry breaking due to quantum fluctuations for positive u can
be characterized by an order parameter
u
ηTMˆ(S)Λˆ η , (3.17)
P α
M≡ 2(1 2α)
− D E
where
Λˆ (1 α)τˆ3+ακˆ3 (3.18)
α
≡ −
and α is a real variational parameter. In terms of Majorana (spin-3/2) components,
1
ηTMˆ(S)Λˆ η =η η +(1 2α)η η . (3.19)
P α 1 4 3 2
2 −
The interaction in Eq. (2.12) can be written as
2
(1 2α) u
H(S) = − d2r + ηTMˆ(S)Λˆ η . (3.20)
I − 2u M 2(1 2α) P α −M
Z (cid:26) (cid:20) − (cid:21)(cid:27)
The interaction is attractive for u > 0 so long as 0 α < 1/2. Precisely for α = 1/2, Eq. (3.19) implies that
≤
2
1ηTMˆ(S)Λˆ η = 0 due to Pauli exclusion (neglecting nontrivial anticommutators). In this case the interaction
2 P α
c(cid:16)annot be writt(cid:17)en as the square of the bilinear.
The zero temperature mean-field condensation energy density is given by
2 1 Λ
∆ ( )=(1 2α)M kdk [ε (k, ) ε (k,0)]+[ε (k, ) ε (k,0)] , (3.21)
1 1 3 3
E M − u − 2π { M − M − }
Z0
where ε (k, ) and ε (k, ) denote the linear and cubic surface band energies modified by the addition of the
1 3
M M
term Λˆ to hˆ in Eq. (2.9); ε (k,0) and ε (k,0) are the unperturbed, gapless linear and cubic dispersion relations
α S 1 3
M
[Eq. (2.10)].
Setting α = 0 (such that Λˆ = τˆ3) and extremizing ∆ ( ) with respect to leads to the mean-field result in
α
E M M
Eq.(2.15), validinthe weakcouplinglimitu c. Since quantumfluctuationsenhance positiveuandthe interaction
≪
is attractive in the Λˆ channel only for 0 α< 1/2, we expect that the order is weighted towards τˆ3 instead of κˆ3;
α
≤
the surface winding number W =+2 throughout this range (Fig. 2).
A curious aspect of Eq. (3.21) is the following. By choosing α arbitrarily close to 1/2, we can suppress the
contribution of the first term on the right-hand side of Eq. (3.21). This allows us to take larger and larger values
for so as to enhance the negativity of the second term. Yet it cannot be that the system wants to condense with
theMbilinear with Λˆ =(τˆ3+κˆ3)/2, since the interaction cannot even be written in terms of its square (as discussed
α
above). This suggests that the true admixture of τˆ3 and κˆ3 order [i.e., the value of α in the expectation value of
M
defined by Eq. (3.17)] cannot be determined by mean-field theory. This warrants further investigation, but we will
not pursue it here.