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Topological phases famously host robust, gapless surface states that reflect the bulk band topology - this is termed the "bulk-boundary correspondence" - and are often the most easily accessible experimental manifestation of the topological physics. In gapped phases such as topological insulators and superconductors, the surface states are energetically separated from the bulk states and can often be treated as independent systems. In topological semimetals, however, both surface and bulk states are gapless and inseparable. We develop theoretical tools that demarcate the surface and bulk contributions to observable phenomena or, alternately, exploit the interplay to derive new physics.
Schematic of surface Luttinger surfaces in (a) topological insulators and (b) Weyl semimetals.
Robust surface Luttinger surfaces
The bulk-boundary correspondence states that topological band structures produce robust surface or edge states are in one-to-one correspondence with the bulk band topology. These states appear as poles in the surface Green's function. We showed that the surface Green's function also contains zeros - whose locus is called a Luttinger surface - that obey the same bulk-boundary correspondence as the surface states. Moreover, the zeros survive even when the surface states are destroyed by symmetry-breaking perturbations on the surface. Learn even more
Surface superconductivity but metallic bulk
According to Landau’s mean-field theory, interacting particles can evade spontaneous ordering far more effectively in lower dimensions. This is why the surface of an ice cube is often watery; the lower dimensionality of the surface impedes the crystallization of H2O molecules, leaving them liquid. We proposed a strange scenario where the opposite seems to occur: the 2D surface of 3D Weyl semimetals enters an ordered phase – a superconductor – while the 3D bulk remains a non-superconducting metal. This picture is consistent with surface-only superconductivity recently seen in t-PtBi2. Learn even more.
Illustration of the parametric regime, shaded gray, where the surface is superconducting while the bulk is in the normal phase. Temperatures above the bulk Tc but below the surface Tc will show surface-only superconductivity.
Closed semiclassical orbits govern the vortex spectrum and endow it with characteristic dependences on the Berry phases and penetration depths of the Fermi arcs, bulk Weyl node locations, vortex orientation and sample thickness.
SUSY and non-local Majorana fermions in superconductor vortices
Superconductor vortices host discrete energy levels that carry critical information about the parent metal. Weyl semimetals have two distinct types of metallic states in their spectrum -- Weyl fermions in the bulk and Fermi arcs on the surface -- that are intricately coupled and cannot be captured by standard Hamiltonians. We ask, "what is the vortex spectrum of a superconductor that descends from a Weyl semimetal?'' We find that the spectrum is determined by semiclassical quantization of cyclotron-like closed orbits consisting of Fermi arcs on opposite surfaces connected by one-way bulk conduits. Miraculously, merely tilting the vortices can transmute them between bosonic, fermionic and supersymmetric, with the last class hosting a pair of non-local Majorana fermions. Thus, we propose a tabletop and tunable realization of a long sought-after system that displays supersymmetry, namely, an equivalence between bosons and fermions. At tilts that we dub "magic angles", the vortex spectrum becomes independent of the slab thickness. In many models and materials, non-local Majorana fermions and supersymmetry exist precisely at these tilts. Learn even more
New route to Majorana fermions
The search for Majorana fermions - particles that are their own anti-particles that were first proposed but never found in high-energy physics - has driven fierce activity in condensed matter, where they occur as zero energy states in topological defects such as vortex cores of type-II superconductors. Among 3D materials, criteria for finding Majorana fermions at the ends of superconducting vortices in time-reversal symmetric insulators and metals are well-known. We derive analogous criteria for the third and last generic type of 3D band structure, namely, time-reversal symmetric Weyl semimetals. The criteria we find depend only on the bulk and surface band structure of the semimetal, which are outcomes of routine photoemission experiments and first-principles simulations, and allow tuning the Majorana fermions in and out of existence by merely tilting an external magnetic field. Learn even more
When the surface Fermi arcs (red curves) plus the projections (black solid lines) of the geodesics between bulk Weyl nodes in the same kz-plane (black dashed lines) form M closed loops, the vortex is gapped and has (lacks) end Majorana modes if M is odd (even). If the Fermi-arc + geodesic-projections form open contours, the vortex is gapless.
Fermi and Luttinger arcs in the first Brillouin zone for different surface terminations in Co3Sn2S2. For a given termination, the two types of arcs form a closed loop since the Brillouin zone is periodic.
Surface Luttinger arcs
Besides the well-known Fermi arcs, we show that the surface of a Weyl semimetal is also endowed with a feature normally associated with strongly interacting systems -- Luttinger arcs, defined as zeros of the electron Green's function. We find that the Luttinger and Fermi arcs form closed loops when the Weyl nodes are undoped. Remarkably, unlike Luttinger contours in strongly interacting systems, the precise shape of the Luttinger arcs here can be determined experimentally by simply peeling off a surface layer. Learn even more
Anomalous surface conductivity
We discover an anomalous contribution to the surface dc conductivity of Weyl semimetals in addition to a normal Drude contribution from the Fermi arc. The anomalous part is independent of the surface scattering time, but involves an effective lifetime due to the non-perturbative coupling of the surface to the bulk. Remarkably, the temperature dependence of the surface conductivity at low temperatures is dominated by the anomalous response which can be probed experimentally. Learn even more
Figure shows that the anomalous conductivity is non-zero when the scattering time (tau) vanishes, while the normal, Drude conductivity shown inset is linear in tau.
In a type-I topological superconductor, the surface Majorana fluid (blue curve) leaks into the bulk beyond the London penetration depth and enhances the bulk penetration of an external magnetic field, depicted by red arrows. The length scales are reversed in a type-II topological superconductor so the enhancement is less striking, but still exists.
Meissner effect in topological superconductors
The expulsion of magnetic fields -- known as the Meissner effect -- is a fundamental property of superconductors. While type-I superconductors expel magnetic fields as long as they are superconducting, type-II superconductors show the Meissner effect below a critical field Hc1. We establish that the surface states of a topological superconductor give rise to an unusual Meissner effect that has a characteristic, power-law temperature dependence. Interestingly, in a type-I superconductor, the magnetic field can "ride on" the surface states and penetrate much deeper into the bulk than the bulk superconductivity would have allowed. Learn even more
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