Guerci et al. presentation

Summary

Superconductivity observed near fractional quantum anomalous Hall states in twisted MoTe2 understandable as a topological chiral superconductor with emergent vortex lattice caused by inhomogeneous emergent magnetic field from moire/layer-pseudospin texture

Significance?

Superconductivity typically evokes attractive pairing or phonon-mediated Cooper p;airing while model produces superconductivity from purely repulsive interactions in flat Chern band → happens near fractional Chern insulator/FQAH physics, where expect strong correlations and Hall order

Experimental motivation

Recent work in twisted MoTe2 shows superconducting signatures around fillings where FQAH behavior appears → explains how two phenomena, fractional Chern insulator physics and superconductivity, arise from same moiré flat-band setting

Ordinary superconductivity?

No: supercond in sense of off-diagonal long range order, phase rigidity, and zero-resistance behavior, but order parameter is unusual, since it’s chiral, topological, and spatially structured into a vortex lattice locked to the moiré lattice

Emergent magnetic field

Not externally applied magnetic field → in twisted bilayer transition metal dichalcogenide (TMD), layer DoF is pseudospin moving through spatially varying moiré texture

Electon moves: texture gives it Berry phase that is equivalent to motion in magnetic field

Paper describes as emergent, spatialyl modulated magnetic field with one flux quantum per moiré unit cell

Compare twisted TMDs to Landau levels

Flat Chern band in moiré material mimics some aspects of a Landau level: nontrivial Berry curvature, Chern number, FQAH-like states

But, emergent field in twisted TMDs is spatially modulated and tied to a lattice, unlike ideal uniform magnetic field in a continuum Landau level

Modulation

Breaks continuous Galilean invariance/continuous magnetic translation symmetry → essential because superconductivity in a uniform magnetic field with Galilean invariance is obstructed, center-of-mass decouples, producing Hall drift and not zero-

Why vortex lattice?

Cooper pair has twice charge of single electron → twice magnetic phase in emergent field

Gauge invariance, magnetic translation symmetry then force superconducting order parameter to wind in space, and the winding manifests as a lattice of vortices

Different from Abrikosov vortex lattice?

In Abrikosov vortex lattice, external magnetic field penetrates conventional type-II superconductor → here vortices arise from emergent magnetic field intrinsic to moiré band structure, not applied magnetic field

Also, vortices pinned to moiré lattice, helping avoid dissipation

Why not dissipation?

Why don’t moving vortices cause dissipation →

moving vortices cause dissipation, but if vortices are strongly pinned, don’t freely move under small currents → authors argue because vortex lattice locked to moiré pattern, zero resistivity can survive below depinning threshold

Double vortices

Double vortices are: superconducting phase winds by more than the usual single-vortex amount per moiré unit cell because Cooper-pair order parameter sees emergent flux with double charge

Describes resulting vortex structure as having double vortices, and these are expected to have gapped bulk quasiparticle spectrum instead of isolated Majorana modes bound to each vortex

Intro slide

Today, we’ll discuss a recent paper entitled “Topological superconductivity with emergent vortex lattice in twisted semiconductors,” by Guerci et al.

The setting is twisted MoTe₂, a moiré material where two layers are rotated relative to each other. That twist creates a long-wavelength moiré superlattice, and in this system the relevant electronic band is not just flat; it is also topological. More specifically, the low-energy band behaves like a Chern band.

That matters because Chern bands are the lattice analog of Landau levels. In an ordinary Landau level, a real magnetic field produces flat bands and quantum Hall physics. Here, the magnetic field is not simply externally applied. Instead, the moiré texture produces an emergent magnetic field through the layer-pseudospin structure of the electrons.

The experimental motivation is that twisted MoTe₂ has shown fractional quantum anomalous Hall physics, and more recently superconducting behavior nearby in filling. That is already surprising. Fractional quantum Hall-like states and superconductors are usually thought of as rather different phases. One is associated with topological order and Hall response; the other with phase coherence and zero resistance.

The question of the paper is: can these two phenomena come from the same microscopic ingredients?

The authors’ answer is yes. Their proposal is that the same emergent magnetic field that helps produce fractional Chern insulating states can also, when spatially modulated, force a superconducting state into a vortex lattice. Even more unusually, this is not an ordinary vortex lattice caused by an external magnetic field. It is an emergent vortex lattice tied to the moiré structure itself.

The punchline is that the superconducting state is chiral, topological, and has a Majorana edge mode. So the paper is not just saying “there is pairing.” It is saying that twisted MoTe₂ may realize a topological superconducting vortex lattice produced by repulsive interactions in a flat Chern band.

A useful roadmap for the talk is this: first, I’ll explain why vortices are forced. Then I’ll describe the microscopic model and what is being diagonalized. Then I’ll explain how the real-space order parameter reveals the vortex lattice. Finally, I’ll explain the topological invariant and why the authors interpret the state as a chiral Majorana superconductor.

Forced vortices slide

The first conceptual point is that the vortices are forced by the emergent magnetic field.

In an ordinary superconductor, the order parameter is a complex field. We can write it schematically as

Delta(r) = |Delta(r)|e^{I theta(r)}

Theta is physically meaningful because superconductivity is a phase-coherent state. 

A vortex is a point where the phase winds by a multiple of (2\pi) around a loop, and the amplitude typically has to vanish at the core.

Usually, when we see vortex lattices in superconductivity, we think of type-II superconductors in an external magnetic field. The external field penetrates the sample in quantized flux tubes, and the superconductor forms an Abrikosov vortex lattice.

Here the logic is comparable, while not identical, the magnetic field being emergent. It comes from Berry phases associated with the layer-pseudospin texture in the moiré band. 

An electron moving through this texture effectively sees a magnetic field. The paper emphasizes that there is one (h/e) flux quantum per moiré unit cell.

Now ask what a Cooper pair sees. A Cooper pair has twice the electron charge. So in the emergent gauge field, the pair effectively sees twice the flux. That means the superconducting order parameter cannot simply be a uniform complex number everywhere in the moiré unit cell. It must pick up phase winding.

This is the basic reason vortices are forced. The superconducting order parameter has to be compatible with magnetic translation symmetry in the presence of emergent flux. A uniform phase would not transform correctly. The way the order parameter satisfies the constraint is by forming a periodic vortex lattice.

A good way to say this in simple terms is: the moiré band geometry acts like a built-in magnetic field, and once electrons pair, the pair wavefunction must accommodate that built-in flux. The accommodation is a lattice of vortices.

There is also an important contrast with ordinary Landau levels. In a perfectly uniform magnetic field with continuous translation symmetry, superconductivity is obstructed in a certain sense: the center-of-mass motion of the pairs drifts rather than giving a stationary zero-resistance state. In this paper, the moiré lattice and the spatial modulation of the emergent field break that continuous symmetry. The vortex lattice is pinned to the moiré pattern, and that pinning is crucial.

And so, the emergent magnetic field does not merely coexist with superconductivity. It determines the spatial structure of the superconducting order parameter.

Microscopic model slide

Let’s take a look at their microscopic model.

The authors do not diagonalize a full atomic-scale Hamiltonian for every orbital in twisted MoTe₂. Instead, they use an effective model projected into the relevant flat Chern band.

This is common in quantum Hall and moiré physics. In the fractional quantum Hall problem, we often project the Coulomb interaction into a Landau level. The kinetic energy is quenched, and the main question becomes: what many-body state is 

selected by interactions within that highly degenerate or nearly flat band?

Here, the analog is a flat moiré Chern band. The Hilbert space consists of electron states in that band. Then the Hamiltonian is essentially an interaction Hamiltonian projected into that band.

Schematically, the projected Hamiltonian has the form

H = P V P

where (P) projects into the Chern band and (V) is the electron-electron repulsion. 

More explicitly, in momentum space, it is a many-body interaction among band-projected creation and annihilation operators. The band geometry enters through form factors: because the Bloch wavefunctions vary across the Brillouin zone, the projected density operators contain information about Berry curvature and quantum geometry.

That is one of the key physical points. Even if the bare interaction is repulsive, after projection into a topological flat band, the interaction is not featureless. It is filtered through the geometry of the band.

So what is being diagonalized?

They take a finite-size system on a torus. They choose a number of moiré unit cells and a number of electrons, corresponding to a filling (\nu). They construct the many-body Hamiltonian in the projected Chern-band Hilbert space. Then they exactly diagonalize that many-body Hamiltonian.

The output is the many-body ground state and low-lying excited states. From those states, they compute diagnostics: whether the state looks like a fractional Chern insulator, whether it has superconducting pair order, whether it has phase stiffness, and what the real-space pair wavefunction looks like.

The important methodological limitation is that exact diagonalization is finite-size. It is powerful because it is unbiased: you are not assuming a superconducting mean-field state from the beginning. But it is limited because you can only study relatively small numbers of particles and unit cells. So the evidence has to come from consistent patterns across several diagnostics, not from one single calculation.

So, they diagonalize a repulsive interaction Hamiltonian after projecting into a flat Chern band. The superconductivity is not inserted by hand. It is inferred from the structure of the resulting many-body ground state.

That is important because the claim is stronger than a mean-field ansatz. The authors are saying that repulsive electrons in this moiré Chern band naturally produce a paired topological state.

RSOP slide

Now we get to the most visually intuitive part: the real-space order parameter.

If the state is superconducting, there should be a dominant pair wavefunction. 

The authors diagnose this using the two-particle reduced density matrix. Conceptually, this is the density matrix for pairs of electrons. If one eigenvalue is much larger than the rest, that means many pairs occupy the same pair wavefunction. That is the superconducting analog of Bose condensation: many particles, or here many pairs, share a common coherent wavefunction.

Once they extract the leading pair wavefunction, they can plot it in real space. This gives the superconducting order parameter across the moiré unit cell.

What do they find? They find that the order parameter is not uniform. Its phase winds, and its amplitude has zeros. Those zeros are vortex cores. The vortices form a pattern locked to the moiré lattice.

This is the concrete confirmation of the earlier symmetry argument. We said that the emergent flux should force vortices. The real-space order parameter shows those vortices.

The paper emphasizes that these are double vortices. The simple way to explain this is that the emergent field contributes one (h/e) flux quantum per moiré unit cell for electrons. But the Cooper pair has charge (2e), so the superconducting order parameter sees twice the effective flux. The phase winding is correspondingly doubled compared to an ordinary single Abrikosov vortex.

One subtlety is that this is not just a picture of an imposed vortex lattice. The order parameter is extracted from the many-body ground state of the projected interacting problem. So the plot is a diagnostic of what the many-body state actually wants to do.

There are three things to highlight when showing the figure.

First, look for zeros in the magnitude of the order parameter. Those are the vortex cores.

Second, look for phase winding around those zeros. That is what makes them vortices rather than just accidental nodes.

Third, notice that the vortex pattern is periodic and tied to the moiré unit cell. That is why the authors call this an emergent vortex lattice.

The takeaway is:

The real-space pair wavefunction turns the abstract claim of superconductivity into a spatial object. It shows a phase-coherent superconducting order parameter whose unavoidable response to the emergent magnetic field is to form a vortex lattice.

Topological invariant slide

Now I’ll turn to the topological invariant, which is the key reason the authors call this a topological superconductor.

In an ordinary band insulator, a filled band can have a Chern number. A nonzero Chern number means there are protected chiral edge modes and a quantized Hall response.

A superconductor is a little different because particle number is not conserved in the same way. The natural description is Bogoliubov–de Gennes theory, where particles and holes are combined into quasiparticles. The BdG Hamiltonian can also have a Chern number, and in a chiral superconductor that Chern number predicts chiral Majorana edge modes.

The paper argues that the superconducting state found in the microscopic model is adiabatically connected to a weak-pairing chiral superconductor. 

That means that, without closing the quasiparticle gap, one can smoothly relate the strongly interacting state to a BdG description where the topology is easier to compute.

The important result is that the superconducting state has Chern number C = -1/2. That is, a chiral complex fermion edge mode has central charge one. 

A chiral Majorana edge mode is half of a complex fermion, so it contributes half as much thermal Hall conductance. Thus the half-integer value is signaling a single chiral Majorana edge mode.

The sign tells us the chirality. In the paper’s figure, the superconducting phase has chirality opposite to the ordinary (C=+1) Chern insulator at full filling. That contrast is important: the same underlying Chern band can produce an ordinary complex-fermion edge mode when filled, but a Majorana edge mode when partially filled and paired.

So the topological invariant is doing two jobs.

First, it establishes that the superconducting state is not topologically trivial.

Second, it predicts an edge structure: a single chiral Majorana mode and a corresponding half-integer thermal Hall response.

There is also a useful caution. The paper does not mean that every vortex necessarily binds a Majorana zero mode. Because the 

vortices here are double vortices, the authors expect the bulk vortex cores to be gapped rather than hosting isolated Majorana zero modes in the usual single-vortex sense. The Majorana physics is primarily in the edge topology.

So the concise version is:

The real-space order parameter shows that the state is a vortex-lattice superconductor. The topological invariant shows that this vortex-lattice superconductor is chiral and supports a Majorana edge mode.

Conclusion slide

We return to the central conceptual message:

The paper gives a unified explanation for two phenomena that might initially seem in tension: fractional quantum anomalous Hall 

physics and superconductivity in twisted MoTe₂.

Both are rooted in the same ingredients: a flat Chern band, strong repulsive interactions, and an emergent magnetic field from the moiré layer-pseudospin texture.

When the emergent field is relatively uniform, it favors fractional Chern insulating physics, analogous to fractional quantum Hall states. But when the emergent field is spatially modulated by the moiré lattice, it can support a superconducting state. That superconducting state cannot be uniform; gauge constraints force it into a vortex lattice. Because the vortex lattice is tied to the moiré pattern, it can be pinned rather than freely drifting.

The microscopic exact diagonalization supports this picture by showing superconducting pair order in the projected interacting Chern band. The real-space pair wavefunction displays the expected vortex lattice. Finally, the topological analysis shows that the state is a chiral topological superconductor with a single Majorana edge mode.

So the broader lesson is that superconductivity in flat topological bands can look very different from textbook BCS superconductivity. It can emerge from repulsion, inherit structure from Berry curvature and moiré geometry, and become topological in a way that naturally connects superconductivity to quantum Hall physics.

In conclusion, we’ve explored how twisted MoTe₂ is not just a platform where superconductivity and fractional quantum Hall physics happen to appear near each other. In this theory, they are two faces of the same moiré Chern-band geometry.