Disorder-enhanced topological protection and universal quantum criticality in a spin-3/2 topological superconductor
We study the Majorana surface states of higher-spin topological superconductors (TSCs) that could be realized in ultracold atomic systems or doped semimetals with spin-orbit coupling. As a paradigmatic example, we consider a model with -wave pairing of spin-3/2 fermions that generalizes . This model has coexisting linear and cubic dispersing Majorana surface bands. We show that these are unstable to interactions, which can generate a spontaneous surface thermal quantum Hall effect (TQHE). By contrast, nonmagnetic quenched disorder induces a surface conformal field theory (CFT) that is stable against weak interactions: topological protection is enhanced by disorder. Gapless surface states of higher-spin TSCs could therefore be robustly realized in solid state systems, where disorder is inevitable. The surface CFT is characterized by universal signatures that depend only on the bulk topological winding number, and include power-law scaling of the density of states, a universal multifractal spectrum of local density of states fluctuations, and a quantized ratio of the longitudinal thermal conductivity divided by temperature . By contrast, for the clean surface without TQHE order would diverge as . Since disorder stabilizes the conducting Majorana surface fluid and quantizes thermal transport, our results suggest a close analogy between bulk TSCs and the integer quantum Hall effect.
The precise quantization of the Hall conductance is the heart of the integer quantum Hall effect (IQHE). The quantization condition holds irrespective of material parameters, quenched disorder, or residual electron-electron interactions. The robustness of the IQHE is due to the tight entwining of the bulk topology (most easily defined for a clean system), Anderson localization (due to quenched disorder, always required in practice to resolve the plateaux), and the chiral edge states whose flow can never be degraded by perturbations such as disorder or interactions, but only redirected. Topological protection in the IQHE locks an easily measured observable directly to a bulk winding number.
In the last decade there has been an explosion of interest in new forms of topological matter, driven by the discoveries of topological insulators and gapless topological phases TIRev2010 (); BernevigHughes2013 (); TSCRev2016 (); TSCRev2016B (). Despite this progress, a three-dimensional analog of the IQHE that ties a robust surface transport signature directly to a bulk winding number remains lacking. One potentially promising route is to look for a generalization of Helium 3B (), the only known bulk topological superfluid predicted to host a gapless surface fluid of unpaired Majorana fermions Volovik2003 (); SRFL2008 ().
Previous theoretical work SRFL2008 (); WZWP2 (); WZWP4 (); YZCP1 (); WZWP3 () has shown that the Majorana surface fluid of a model spin-1/2 bulk topological superconductor (TSC) can be robust to both disorder and interaction effects, and should exhibit a universal surface thermal conductivity proportional to the bulk winding number WZWP3 (). In all of these works, the form of the 2D surface theory was always assumed to be relativistic, with “colors” of linearly-dispersing Majorana fermions coupled via interactions and/or quenched disorder; denotes the winding number. A key unanswered question is whether the physics (e.g., universal thermal conductivity) is tied to this simplifying assumption, or instead represents a robust aspect of generic bulk TSCs.
Recent theoretical work has turned to higher-spin TSCs, with potential applications to alkaline and alkaline-earth ultracold atoms Wu2016 () or doped semimetals with spin-orbit coupling Fang2015 (); Paglione2016 (); Brydon2016 (). In this paper, we consider the surface states of a spin-3/2 generalization of with isotropic -wave pairing Wu2016 (). A novel feature is that the surface Majorana fluid exhibits coexisting linear and cubic bands. Cubic surface bands were also predicted in a closely related model Fang2015 () that may be relevant for superconducting half-Heusler alloys. Due to the van Hove singularity, one might expect that any residual interactions between surface Majorana particles would produce a strong instability. Surprisingly, we show that interactions are only marginally relevant: only attractive interactions induce spontaneous time-reversal symmetry breaking and lead to a surface thermal quantum Hall effect (TQHE) SRFL2008 (); WZWP2 (). This weak instability is tied to the strong constraints imparted by Pauli exclusion to a Majorana gas, despite the density of states divergence. Repulsive interactions are marginally irrelevant; their main effect would be to generate a finite longitudinal surface thermal conductivity at temperature due to inelastic scattering. In the absence of impurity scattering the ratio would diverge as .
By contrast, nonmagnetic quenched disorder proves to be a strong perturbation. Using exact diagonalization to study the noninteracting dirty surface, we show that disorder induces scaling consistent with a critical, exactly solvable conformal field theory (CFT) SO WZWP4 (). (Here denotes the number of replicas.) The CFT governs the divergence of the global density of states and the statistics of the single-particle wave functions. The level of the current algebra (=4) is also the modulus of the bulk winding number for our -wave model. This is the same result that obtains for spin-1/2 models of TSC surface states studied previously WZWP4 (); WZWP3 (). In the spin-1/2 case with winding number , the clean surface fluid is a free fermion (level one) CFT due to the relativistic dispersion. The emergence of another CFT with level in the presence of disorder follows from certain rules in these theories (conformal embeddings WZWP4 (); WZWP3 ()).
Here the situation is very different. The clean Majorana fluid of the spin-3/2 model is not a CFT, as evidenced by the cubic dispersion. Moreover, a standard derivation of the effective surface theory with disorder would incorrectly predict a thermal metal with weak antilocalization SenthilFisher2000 (); WZWP4 (). Properties of this metal would depend on the bare disorder strength and would vary slowly with system size. Our numerics instead show universal 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 empirically suggest a deep relation between the topology of the bulk and the CFT describing the disordered surface of a TSC, despite the fact that the clean surface theories can fundamentally differ. Technically, it means that the topology precisely tunes the surface field theory to the conformal fixed point made possible by a Wess-Zumino-Novikov-Witten term WZWP4 (). Without this fine-tuning, this fixed point is unstable to the thermal metal phase (despite the WZNW term) WZWP3 ().
The SO CFT that describes the disordered Majorana surface fluid is known to be protected against weak interaction effects WZWP4 (). We conclude that disorder stabilizes the surface Majorana fluid for this spin-3/2 model, and this implies that higher-spin TSCs could be robustly protected. The thermal Hall conductivity divided by temperature of the surface TQHE is quantized and universal KaneFisher (); ReadGreen2000 (); Capelli02 (); Zhang2011 (); Ryu2012 (); Stone2012 (): , where is the surface winding number and
for the disorder-induced surface CFT (which preserves time-reversal symmetry). Here the winding number . Eq. (2) implies that the low-temperature thermal conductivity is quantized by the bulk winding number, independent of both disorder and interactions. Since disorder stabilizes the surface and induces a quantized thermal conductivity, bulk TSCs appear to be closely analogous to the integer quantum Hall effect in two dimensions. Our results are summarized by the phase diagram in Fig. 1.
This paper is organized as follows. In Sec. II, we define the bulk model and describe the form of the surface states. We then summarize our results regarding the marginal instability of the clean surface, and the universal quantum criticality of the disordered one. The rest of the paper explains key technical details. Sec. III shows the derivation of the surface state Hamiltonian and the calculation of the surface winding number in the presence of explicit time-reversal symmetry breaking. The effects of interactions on the clean surface are treated using one-loop renormalization and mean field theory. Sec. IV describes the incorporation of disorder, and provides details of the numerical diagonalization scheme.
Ii Model and results
ii.1 Bulk and surface models
We consider a system of spin-3/2 fermions. In the absence of pairing, we assume a bulk Hamiltonian of the form
where . The 4-component fermion field has -label ; are spin-3/2 operators. Eq. (3) is an isotropic version of the Luttinger Hamiltonian YuCardona () used to model heavy and light hole bands in zinc-blende semiconductors; the parameter measures the strength of effective spin-orbit coupling (SOC) amongst the states of the 3/2 multiplet. Here we have set in the first term ( is the band mass in the absence of SOC). We assume that , so that both bands “bend up.” The situation where bands bend oppositely is relevant for half-Heusler alloys; in that case similar Majorana surface states can arise with bulk -wave pairing Fang2015 (), but the winding numbers differ Ghorashi2017 ().
We assume isotropic -wave pairing of spin-3/2 fermions Wu2016 ():
where the Bogoliubov-de Gennes Hamiltonian is
Here is the chemical potential and the BCS gap parameter. The 8-component field in Eq. (4) has the particle-hole space decomposition
The field satisfies the “Majorana” condition , where . This implies the automatic particle-hole symmetry Time-reversal invariance is encoded as where . Combining particle-hole and time-reversal Wu2016 (); SRFL2008 () gives the effective chiral condition With all of these symmetries the model belongs to class DIII SRFL2008 (). The bulk winding number is Wu2016 () so long as and .
The effective surface Hamiltonian obtains by terminating the system in the -direction and diagonalizing . The momentum labels propagation parallel to the surface. For and hard wall boundary conditions, there are four zero energy bound states . The most important feature is that the particle-hole “spin” locks to the physical spin at the surface:
The particle-hole spin points along the () direction for positive (negative) . In Eq. (8), denotes the bound state envelope function.
Since time-reversal gives the effective chiral condition in the bulk, the locking condition implies that the effective surface Hamiltonian satisfies
where () for (). We introduce two mutually commuting species of Pauli matrices: anticommute with and act on the space, while mix the and states with the same sign. E.g., The matrix . Then the locking condition implies the automatic surface particle-hole symmetry
Here, , , , the coefficient for the linear dispersion is normalized to one, and is a real constant with units of length. For weak SOC (), it is easy to see that . Without SOC, second-order kp theory gives so that the bands remain flat. Nonzero is symmetry-allowed and thus expected in the generic situation; an alternative route incorporates additional small -wave pairing. In Eq. (II.1), we neglect terms cubic in because these do not modify the low-energy dispersion relations.
ii.2 Surface perturbations
The Hamiltonian for the surface fermion fluid is given by
where is a four-component Majorana spinor and is the position vector. Local bilinear (“potential”) perturbations must obey surface particle-hole symmetry. There are 6 Hermitian terms without derivatives of the form , where or satisfies Eq. (10). These are classified by symmetry.
Under planar rotations, and are scalars, transform like a vector, and transform like second-rank tensor components. Only are time-reversal even [Eq. (9)]; the rest are odd.
A generic combination breaks time-reversal and induces a gapped surface thermal quantum Hall (TQH) state ReadGreen2000 (); TSCRev2016 (). We compute the surface winding number using the Green’s function Volovik2003 (); the result is shown in Fig. 2. The lines are gapless surface plateau transitions. The maximum winding number and surface gap for fixed is achieved for , i.e. order. Finally, we note that the spin operator corresponds to , so that an external Zeeman field would induce the plateaux.
ii.3 Marginal instability of the clean surface
Although we treat the gapped bulk as an effectively non-interacting mean field Hamiltonian, we must consider the effects of residual interactions on the surface Majorana fluid carefully. This is because the latter is 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 residual interactions can be mediated by virtual fluctuations of the “massive” electromagnetic field. Here we posit the form of the interactions based on symmetry and Pauli exclusion.
Because is a four-component Majorana field, there is only a single interaction term without derivatives that we can write; others with derivatives are less relevant. Labeling the components as ,
where the minus (plus) sign corresponds to (). Thus is an attractive (repulsive) interaction in the () channel. The coupling has units of length. The sign of could be determined by integrating out the bulk superfluid and the electromagnetic field, but we will not do so here.
Given that the noninteracting surface fluid has coexisting linear and cubic bands, it is not a priori obvious how to assess the relevance of from a renormalization group (RG) perspective. Moreover, the van Hove singularity suggests that nonzero will induce bad infrared behavior. In fact one-loop perturbation theory gives the simple vertex correction,
where is the ultraviolet momentum cutoff. The correction is only logarithmic, and is cut in the infrared by the length scale . This immediately implies the beta function,
where is the dimensionless coupling.
The absence of bad infrared behavior in Eq. (15) and the weakness of the ultraviolet singularity is due to Pauli exclusion, i.e. the fact that both linear and cubic components of the Majorana Green’s function must appear simultaneously in the loop. Eq. (16) implies that is a marginally relevant perturbation. Eq. (II.3) suggests a natural interpretation in terms of TQH order with surface winding number .
We can confirm this picture with a mean-field calculation. We decouple the interaction in Eq. (II.3) with the order parameter . Zero-temperature mean-field theory gives
Physically we can associate nonzero with surface “i ” (imaginary -wave) pairing of the Majorana particles WZWP2 (); WZWP4 (); Nomura2015 (). This can be understood via the following argument. In the bulk, one can write a local spin singlet, time-reversal odd pairing operator
Eq. (8) implies that this bilinear projects to at the surface. On the other hand, Eq. (II.3) can also be written as proportional to , implying surface magnetic order for . Indeed, these disparate orders are unified in the surface fluid, due to the strong spin-orbit coupling in the bulk and the locking condition [Eq. (8)]. We might anticipate a generic order parameter of the form [c.f. Fig. 2] with . Although we are confident that the surface resides in the or plateau for , there are hints that mean-field theory fails to correctly predict the admixture of and . For details, see Sec. III.4, below.
ii.4 Quenched disorder and universal surface quantum criticality
In a solid state realization, quenched disorder due to impurities and other defects is inevitable at the sample surface. Now we consider the effects of disorder on the non-interacting surface states.
We add real disorder potentials that couple to the time-reversal symmetric bilinear perturbations to Eq. (II.1):
where and . We consider only time-reversal invariant disorder; equivalently, we require that there are no magnetic fields or magnetic impurities at the surface. We assume Gaussian white noise disorder potentials with common variance given by a dimensionless parameter . Due to the cubic dispersion, even weak disorder is expected to produce a strong effect. Indeed, perturbative renormalization of produces a quadratic infrared divergence, and implies that is the effective disorder strength. This has dimension 2 and thus corresponds to a strongly relevant perturbation of the clean surface band structure. To treat the disorder nonperturbatively, we diagonalize Eq. (19) numerically. The calculation is performed in momentum space to avoid fermion doubling issues YZCP1 (). Details are described in Sec. IV.2, below.
In spin-1/2 bulk TSCs with relativistic surface fluids, conformal embedding rules establish certain 2+0-D conformal field theories (CFTs) as governing the properties of disordered, noninteracting surface states WZWP4 (). For a class DIII bulk with winding number 4, that theory would predict the surface CFT SO, where counts replicas. We will demonstrate that this theory also governs the dirty surface states of the spin-3/2 TSC.
The SO theory predicts WZWP4 () a diverging low-energy global density of states (DoS) . Note that this is a weaker power law than the 1/3 van Hove singularity in the clean system. In Fig. 3, we compare numerical results for different disorder strengths to the CFT prediction for the integrated DoS . We find good agreement irrespective of the disorder strength.
The disorder-induced spatial fluctuations of the critical surface wave functions are encoded in the multifractal spectrum . (For a recent review on multifractality at Anderson metal-insulator transitions, see e.g. Ref. AndRev08 ().) The SO theory predicts an exactly quadratic spectrum WZWP4 () for the low-energy wavefunctions,
Figs. 3 and 4 provide strong evidence that the disordered, noninteracting spin-3/2 Majorana fluid is governed by the SO theory. This is surprising because a standard derivation SenthilFisher2000 () of the disorder-induced effective field theory would predict a thermal metal phase exhibiting weak antilocalization. Although the theory in SenthilFisher2000 () should be augmented by a Wess-Zumino-Novikov-Witten term (see Sec. IV.1 for details), this term does not alter the tendency towards antilocalization in the metallic phase WZWP4 (). Because the clean density of states diverges for the surface Majorana fluid studied here, a diffusive metallic state would be generically expected. Yet this is inconsistent with our numerical results [which instead match the SO CFT]. An important technical point is that the CFT is unstable to the thermal metal phase [see Eq. (43)]. It means that the CFT can only be realized if the system is tuned to the SO fixed point. Our numerical results imply that this is exactly what happens.
The same “fine-tuning” is required for the spin-1/2 TSCs. In that case, however, there is a nonperturbative argument for it using conformal embedding theory WZWP4 (). Additional evidence in the spin-1/2 case obtains by comparing interaction (Altshuler-Aronov) corrections to transport via two methodologies: (1) order-by-order in the interaction strength, in a fixed realization of disorder, and (2) within a disorder-averaged large winding number expansion. These give the same result only if the disorder-averaged system is tuned to the CFT (in which case Altshuler-Aronov corrections vanish) WZWP3 (). We do not have the conformal embedding dictionary WZWP4 () utilized for spin-1/2 TSCs, but our numerical results empirically suggest that there is an equivalence between the bulk topology and the CFT describing the disordered surface of a TSC WZWP4 (); YZCP1 (); WZWP3 (), despite the fact that the clean surface theories can fundamentally differ.
ii.5 Stability, phase diagram, and quantized thermal conductivity
We have seen that the clean surface is marginally unstable to TQH order. We have also shown that quenched disorder is a strong perturbation that drives the non-interacting surface to a phase described by the SO CFT. It is known WZWP4 () that interactions are strongly irrelevant to this CFT,
where is the dimensionless coupling strength. Although multifractality can sometimes enhance interactions Feigelman07 (); Feigelman10 (); WZWP2 (), that does not occur here. The reason is again Pauli exclusion: the interaction and second multifractal moment operators are distinct due to the complete antisymmetrization of the former WZWP4 (). We therefore conclude that disorder stabilizes the surface Majorana fluid of this spin-3/2 TSC. This is our most important result.
We note that Eq. (21) technically obtains from the dynamical version of the SO theory. This is a 2+1-D theory of Majorana fermions propagating in space and time, whose disorder-averaged spatial correlations and dynamical scaling exponent are governed by the 2+0-D replicated CFT, see WZWP4 () for details. Thus in Fig. 1, “SO” really refers to this dynamical hybrid theory, which can also be expressed as a Wess-Zumino-Novikov-Witten Finkel’stein nonlinear sigma model (WZNW-FNLsM) WZWP4 (); WZWP3 ().
The thermal conductivity of the WZNW-FNLsM receives no quantum interference corrections due to disorder SRFL2008 (); WZWP3 () at the conformal fixed point. Interaction-mediated Altshuler-Aronov corrections also vanish to at least order 1/, where is the bulk winding number. The absence of Friedel oscillations in any Majorana surface “density” implies that these should be absent to all orders WZWP3 (). Since the spin-3/2 Majorana surface fluid studied here realizes the SO theory in the presence of disorder, we conclude that the ratio of the longitudinal thermal conductivity and temperature is precisely quantized as [Eq. (2)].
In summary, we have derived surface states and surface effective Hamiltonian for a spin-3/2 time-reversal invariant topological superconductor that hosts a cubic dispersion coexisting with the conventional linear Majorana cone. We have shown that in the clean limit, unlike the spin-1/2 case Maciejko15 (), interactions are marginally relevant and lead to a BCS-type instability that gaps out the surface and induces a thermal quantum Hall effect (TQHE) plateau. By contrast, quenched disorder gives the SO theory previously predicted for a spin-1/2 TSC with winding number ; this theory is stable to interaction effects. We conclude that disorder enhances topological protection. In the low-temperature limit, the ratio of the longitudinal thermal conductivity to temperature is predicted to be quantized and proportional to the bulk winding number, as shown in Eq. (2).
Iii Ideal (clean) surface states and interactions
iii.1 Derivation of the surface Hamiltonian Eq. (ii.1)
iii.1.1 Luttinger Hamiltonian bulk
To obtain the surface Hamiltonian, we consider a bulk superconductor in the half space with hard wall boundary conditions. We divide Eq. (5) into two parts:
where is the momentum parallel to the surface, , and are spin-3/2 raising and lowering operators. In Eq. (22a), we have added a time-reversal symmetric -wave pairing term proportional to . We will utilize this below in the case of vanishing bulk spin-orbit coupling (SOC). Energies like and have units of 1/(length), while has units of 1/length.
In this subsection we will ignore -wave pairing () and we will treat the SOC term proportional to as a small perturbation (although this is not necessary). The surface eigenstates of with zero transverse momentum satisfy and take the form shown in Eq. (8),
Here the explicit 2-component spinor resides in particle-hole () space; the four zero energy states are distinguished by their eigenvalues . The particle-hole spinor is “locked” to the physical spin, as it points along the () direction for positive (negative) . The form of the envelope function is appropriate for the weak pairing limit, .
To obtain the effective surface Hamiltonian for nonzero transverse momentum, we diagonalize in the basis of zero modes given by Eq. (23). The only non-vanishing elements obtain from
iii.1.2 Vanishing SOC in the bulk
If the SOC parameter , then the surface bands remain flat in degenerate perturbation theory. Another way to get nonzero in Eq. (II.1) is by incorporating an additional weak -wave pairing amplitude , as in Eq. (22a). Then the zero energy surface eigenstates with vanishing transverse momentum again take the form shown in Eq. (8), but with the modified envelope function
Here we assume that (so as to remain in the bulk topological phase with winding number Wu2016 ()). For convenience, we also assume intermediate strength pairing such that ; in this case there is only one branch of bulk scattering states.
To obtain the effective surface Hamiltonian for nonzero transverse momentum, we use kp theory. The matrix elements of [Eq. (22b) with ] give the -linear terms in Eq. (II.1), which connect the states. To connect the states to the former, one has to go to second order. This yields the matrix elements Sakurai ()
where projects out of the degenerate zero mode space. Eq. (26) can be expressed using the basis of bulk scattering states with zero transverse momentum:
where denotes a scattering state with standing wave momentum (oscillation in the -direction), -eigenvalue , and gapped positive energy eigenvalue , while is the (negative energy) particle-hole conjugate of . The matrix was introduced in Eq. (7).
The scattering states take the form
where and are 2-component spinors in particle-hole space. The expressions for these and are unwieldy so we omit them here.
Finally, one computes Eq. (27) using Eqs. (8), (25), and (28). This second-order result vanishes for . Nonzero in Eq. (II.1) is symmetry allowed, and thus expected in the generic situation. The simplest way to get it is by retaining nonzero in the bound states , but neglecting it in the scattering states (which become very complicated for ). This gives nonzero terms mixing the and states proportional to and in (above and below the diagonal, respectively, consistent with planar rotational invariance). Without loss of generality, we can take the coefficients to be real and positive since the phases can be removed with a unitary transformation.
iii.2 Calculation of the surface winding number in Fig. 2
We compute the surface winding number for the clean, noninteracting Majorana surface fluid perturbed by time-reversal breaking “mass” terms. The Hamiltonian is
where was defined by Eq. (II.1). The energy bands of are gapped for non-zero values of unless , in which case a gapless linear Dirac point appears at .
Since the mass terms break surface time-reversal symmetry [Eq. (9)], the surface theory resides in class D SRFL2008 (). In 2D, this class can exhibit a thermal quantum Hall effect KaneFisher (); ReadGreen2000 (); Capelli02 (), where edge states carry a quantized energy current. The thermal Hall conductivity can be expressed in terms of a winding number via Zhang2011 (); TSCRev2016 ()
where was defined by Eq. (1). In terms of the surface Green’s function
the winding number is given by Volovik2003 ()
iii.3 Perturbative vertex renormalization
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
valid in the limit of vanishing external frequencies and momenta. The three double Levi-Civita terms in the square brackets correspond to the three loop corrections shown in Fig. 5. We emphasize that the sign of each diagram has to be carefully determined using Wick’s theorem for the Majorana fermion field. The Green’s function is given by Eq. (31) with , while was defined by Eq. (10).
iii.4 Mean-field theory: surface thermal quantum Hall plateaux
The interaction strength is enhanced (suppressed) by quantum fluctuations for () [Eq. (16)]. In Eq. (II.3) and the text following, it is noted that is an attractive (repulsive) interaction in the “” (“”) 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 can be characterized by an order parameter
and is a real variational parameter. In terms of Majorana (spin-3/2) components,
The interaction in Eq. (II.3) can be written as
The interaction is attractive for so long as . Precisely for , Eq. (40) implies that due to Pauli exclusion (neglecting nontrivial anticommutators). In this case the interaction cannot be written as the square of the bilinear.
The zero temperature mean-field condensation energy density is given by
where and denote the linear and cubic surface band energies modified by the addition of the term to