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

The gist: Disorder stabilizes higher-spin topological superconductors by inducing a surface conformal field theory that quantizes thermal transport, suggesting a close analogy between bulk topological superconductors and the integer quantum Hall effect.

Model and Surface States

The study considers a model with p-wave pairing of spin-3/2 fermions that generalizes 3He-B, which has coexisting linear and cubic dispersing Majorana surface bands The bulk Hamiltonian is defined by an isotropic version of the Luttinger Hamiltonian used to model heavy and light hole bands in zinc-blende semiconductors <ref:1701.02787#pg4> The effective surface Hamiltonian obtains by terminating the system in the z-direction and diagonalizing hˆ(kx, ky, −i∂z) <ref:1701.02787#pg4>. For k = 0 and hard wall boundary conditions, there are four zero energy bound states ψ0,msi> <ref:1701.02787#pg4>. The most important feature is that the particle-hole “spin” locks to the physical spin at the surface ψ0,msi> = σ1 = sgn(ms)i ⊗ msi ⊗ fmsi> <ref:1701.02787#pg4>.

Surface Perturbations and Instabilities

The Hamiltonian for the surface fermion fluid is given by H(S)0 = 1/2 Z d2r ηT Mˆ (S)P hˆS η <ref:1701.02787#pg5>. There are 6 Hermitian terms without derivatives of the form 1/2 R d2r ηT Mˆ (S)P Λˆ η, where Λ ∈ τ1,2 or κ1,23 satisfies Eq. (2.8) <ref:1701.02787#pg5>. A generic combination Λ =ˆ m1 τ3 +m2 κ3 breaks timereversal and induces a gapped surface thermal quantum Hall (TQH) state <ref:1701.02787#pg5>. The lines m1 = ±m2 are gapless surface plateau transitions <ref:1701.02787#pg3>.

Marginal Instability of the Clean Surface

Although the bulk is treated as an effectively non-interacting mean field Hamiltonian, residual interactions on the surface must be considered <ref:1701.02787#pg4>. Because η 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 <ref:1701.02787#pg5>. One-loop perturbation theory gives the simple vertex correction Γ(4) = -u + u 2/4πc ln(4cΛ) <ref:1701.02787#pg4>. This immediately implies the beta function, du˜dl = u˜2/4π + O u˜3 <ref:1701.02787#pg4>. The absence of bad infrared behavior in Eq. (2.13) 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 <ref:1701.02787#pg5>.

Disorder-Induced Universal Quantum Criticality

Quenched disorder proves to be a strong perturbation that drives the noninteracting surface into a phase described by a critical, exactly solvable conformal field theory (CFT) SO(n)4 <ref:1701.02787#pg4>. The CFT governs the divergence of the global density of states and the statistics of the single-particle wave functions <ref:1701.02787#pg4>. In spin-1/2 bulk TSCs with relativistic surface fluids, conformal embedding rules establish certain 2+0-D CFTs as governing the properties of disordered, noninteracting surface states <ref:1701.02787#pg4>. The SO(n)4 theory predicts a diverging low-energy global density of states (DoS) ν(ε) ∼ ε-1/5 <ref:1701.02787#pg4>. Numerical results in Figs. 3 and 4 instead imply universal behavior in the disorder-averaged DoS and multifractal spectrum for the spin-3/2 surface Majorana fluid, consistent with the SO(n)4 CFT <ref:1701.02787#pg4>.

Stability and Quantized Thermal Conductivity

The interaction strength u is enhanced (suppressed) by quantum fluctuations for u > 0 (u. The thermal conductivity of the WZNW-FNLsM receives no quantum interference corrections due to disorder at the conformal fixed point <ref:1701.02787#pg6>. The absence of Friedel oscillations in any Majorana surface “density” implies that these should be absent to all orders <ref:1701.02787#pg6>. Since the spin-3/2 Majorana surface fluid studied here realizes the SO(n)4 theory in the presence of disorder, we conclude that the ratio of the longitudinal thermal conductivity and temperature is precisely quantized as T → 0 <ref:1701.02787#pg8>.

Summary

Disorder enhances topological protection by stabilizing the surface Majorana fluid for this spin-3/2 TSC <ref:1701.02787#pg6>. The low-temperature limit predicts that the ratio of the longitudinal thermal conductivity to temperature is quantized and proportional to the bulk winding number, as shown in Eq. (1.2) <ref:1701.02787#pg6>. This quantization remains true even when interaction (Altshuler-Aronov) corrections are taken into account <ref:1701.02787#pg6>. The results suggest that bulk TSCs appear to be closely analogous to the integer quantum Hall effect in two dimensions <ref:1701.02787#pg6>.

How it works

1. In the clean limit, 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 <ref:1701.02787#pg6>.

2. Quenched disorder gives the SO(n)4 theory previously predicted for a spin-1/2 TSC with winding number 4; this theory is stable to interaction effects <ref:1701.02787#pg4>.

3. The disorder-induced spatial fluctuations of the critical N(ε)/N(ε∗) ε/ε∗ λ = πλ = 4π/5λ = 3π/5λ = 2π/5λ = π/5 <ref:1701.02787#pg5>.

4. The CFT is unstable to the thermal metal phase <ref:1701.02787#pg4>, meaning the CFT can only be realized if the system is tuned to the SO(n)4 fixed point <ref:1701.02787#pg4>.

5. The coupling strength is pinned to a special value equal to the winding number 4, times a universal constant, which implies that the topology “fine tunes” λ to its fixed point value <ref:1701.02787#pg4>.

Quenched Disorder

The standard theory for a disordered class DIII system suggests that thermal conductance is determined by the bare strength of the disorder G ∝ 1/λ, and G grows with increasing system size due to weak antilocalization <ref:1701.02787#pg13>. Our numerical results instead imply universal behavior in the disorder-averaged DoS and multifractal spectrum for the spin-3/2 surface Majorana fluid, consistent with the SO(n)4 conformal field theory <ref:1701.02787#pg4>. The CFT has a sigma model description that is almost identical to the one obtained by the “standard program,” but augmented with a Wess-Zumino-Novikov-Witten term <ref:1701.02787#pg4>. The key point is that the coupling strength is pinned to a special value equal to the winding number 4, times a universal constant <ref:1701.02787#pg4>.

Momentum Space Exact Diagonalization

For exact diagonalization, we choose a momentum cutoff Nk and keep only those points n in momentum space with −Nk ≤ ni ≤ Nk, for i = 1, 2 <ref:1701.02787#pg13>. This choice corresponds to an energy cutoff of Λ = 2πNk/L <ref:1701.02787#pg13>. The resulting approximate Hamiltonian [hˆ(P)S]m,n is a dense numerical matrix acting on a 4(2Nk + 1)2-dimensional Hilbert space that we diagonalize to obtain the energy values and eigenstate wave functions. The results are robust with respect to variation of the system size Nk, the correlation length ξ, and c.

Stability, Phase Diagram, and Quantized Thermal Conductivity

We have seen that the clean surface is marginally unstable to TQH order <ref:1701.02787#pg6>. We have also shown that quenched disorder is a strong perturbation that drives the noninteracting surface to a phase described by the SO(n)4 CFT <ref:1701.02787#pg6>. It is known that interactions are strongly irrelevant to this CFT, du˜dl = -u˜2 + O u˜2 <ref:1701.02787#pg6>. We therefore conclude that disorder stabilizes the surface Majorana fluid of this spin-3/2 TSC <ref:1701.02787#pg6>. The thermal conductivity of the WZNW-FNLsM receives no quantum interference corrections due to disorder at the conformal fixed point <ref:1701.02787#pg6>. The absence of Friedel oscillations in any Majorana surface “density” implies that these should be absent to all orders <ref:1701.02787#pg6>.

Final Conclusion

We conclude that disorder enhances topological protection <ref:1701.02787#pg6>. 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. (1.2) <ref:1701.02787#pg6>. This quantization remains true even when interaction (Altshuler-Aronov) corrections are taken into account <ref:1701.02787#pg6>.

Acknowledgements

We thank Bitan Roy for helpful discussions.

1. Improved AI systems can predict robust topological protection in solid-state systems by mapping bulk winding numbers to quantized thermal transport ratios, as suggested by the ratio of the longitudinal thermal conductivity and temperature is precisely quantized as T → 0 [Eq. (1.2)] for the disorder-induced surface CFT.

2. AI systems can model universal quantum criticality in disordered topological superconductors by incorporating the SO(n)4 theory that governs the dirty surface states, allowing prediction of universal scaling that is independent of the disorder strength.

3. The improved system can distinguish between interaction effects and disorder effects on surface physics by predicting that disorder stabilizes the surface Majorana fluid for this spin-3/2 model, contrasting this with interactions that lead to a TQHE order when attractive interactions induce spontaneous time-reversal symmetry breaking.

4. AI systems can perform nonperturbative calculations of the interaction strength in the presence of disorder, predicting that disorder enhances topological protection because interactions are strongly irrelevant to this CFT, as indicated by the renormalization group result du˜dl = −u˜2 + O u˜2.