Theory of reentrant superconductivity in Corbino Josephson junctions

This paper theoretically demonstrates that non-circular Corbino Josephson junctions on topological insulators exhibit reentrant superconductivity with a period halved in the topological case compared to the trivial one, offering a potential signature for detecting topological superconductivity.

The Gist

The Big Picture: A Race Around a Track

Imagine you have a race track made of a special material that allows electricity to flow without any resistance (superconductivity). Usually, if you put a magnet near this track, it messes up the flow, and the electricity stops. This is like a standard "Josephson junction."

But the researchers in this paper are looking at a specific shape: a Corbino junction. Instead of a straight track, imagine a donut. There is an inner ring and an outer ring of superconducting material, and the space in between is filled with a "normal" metal (or a special topological material).

They are asking: What happens to the super-current if we thread a magnetic field through the hole in the middle of the donut?

The Standard Rule: The "Fraunhofer" Pattern

In a normal, straight superconducting wire, if you increase the magnetic field, the current goes up and down in a wave pattern (like a heartbeat). It hits zero at specific points. This is called the Fraunhofer pattern.

In a circular donut-shaped junction, the rules are strict. The magnetic field has to come in "chunks" (quantized). The paper says that for a perfectly circular donut, as soon as you add even one chunk of magnetic field, the super-current dies completely. It's like a race where the moment a runner stumbles, the whole team is disqualified.

The Twist: Shape Matters (The "Square" Donut)

The researchers realized that real-world donuts aren't always perfect circles. What if the donut is shaped like a square?

They found something surprising:

• In a normal square donut: The super-current doesn't just die when you add magnetic field. It comes back to life!
• The "Reentrant" Effect: Imagine the current is a light that turns off when you add a little magnet. But if you keep adding magnets in specific amounts, the light turns back on. This is called "reentrant superconductivity."
• The Corner Rule: The light only turns back on if the number of magnetic chunks matches the number of corners. For a square (4 corners), the current only returns when you have 4, 8, 12 chunks of magnetism. It's like a lock that only opens if you turn the key a specific number of times based on how many corners the shape has.

The Magic Material: Topological Insulators

Now, the researchers swapped the "normal metal" in the donut for a Topological Insulator.

• Analogy: Think of a normal metal as a busy highway where cars (electrons) can crash into each other. A topological insulator is like a magic highway where cars are forced to drive in a single file line and cannot crash or turn back. They are "protected" by the laws of physics.
• These special highways have "chiral Majorana modes," which are like ghostly runners that can only go one way.

The Discovery: Halving the Period

When they put this "magic highway" material into the square donut, the rules changed again.

• Normal Square: The current only comes back on for multiples of 4 (4, 8, 12...).
• Topological Square: The current comes back on for multiples of 2 (2, 4, 6, 8...).

The "Period Halving":
Imagine you are clapping to a beat.

• In the normal square, you clap every 4 beats.
• In the topological square, you clap every 2 beats.

The "beat" (the pattern of when the current returns) has been cut in half. The paper suggests that if you see this "halving" effect in an experiment, it is a strong sign that you have created a topological superconductor. It's a fingerprint that proves the material is doing something exotic.

Why This Matters (According to the Paper)

The authors say this is a new way to test for topological superconductivity.

1. Geometry is Key: You don't need a perfect circle. In fact, using a shape with corners (like a square) makes the effect much easier to see.
2. A Simple Test: By counting how many times the current turns back on as you increase the magnet, you can tell if the material is "normal" or "topological."
3. The "Diode" Effect: They also found that if the shape isn't perfectly symmetrical, the current might flow better in one direction than the other, switching back and forth as you change the magnet. This is like a traffic light that changes color depending on how many cars are waiting.

Summary

The paper calculates that if you build a donut-shaped superconducting junction with corners:

• Normal materials: The current returns only when the magnetic field matches the number of corners.
• Topological materials: The current returns twice as often (half the distance).

This "period halving" is a unique signature that could help scientists prove they have successfully built a topological superconductor, a material that could be very useful for future quantum computers (though the paper focuses on the detection method, not the computer building itself).

Technical Summary

Technical Summary: Theory of Reentrant Superconductivity in Corbino Josephson Junctions

Problem Statement
Josephson junctions (JJs) formed by conventional superconductors typically exhibit Fraunhofer-like oscillations in their critical current ($I_c$) as a function of magnetic flux ($\Phi$). When these junctions are placed on the surface of three-dimensional topological insulators (3DTIs), the presence of chiral Majorana modes is expected to modify this pattern. While previous studies on planar JJs on 3DTIs have suggested "node lifting" (where $I_c$ does not reach zero at integer fluxes), these deviations are often difficult to discern experimentally. Furthermore, planar geometries do not enforce fluxoid quantization, limiting the direct probing of the current-phase relation (CPR). This paper addresses the need for a more robust theoretical framework to distinguish between topologically trivial and non-trivial superconductivity by analyzing Corbino Josephson junctions, where the geometry is closed and fluxoid quantization is strictly enforced.

Methodology
The authors employ a combination of analytical modeling and numerical simulation to investigate the critical current of Corbino JJs with varying geometries (circular vs. non-circular) and material compositions (conventional metal vs. 3DTI surface states).

1. Geometric Modeling: The junction is defined by inner and outer boundaries $r_{in}(\theta)$ and $r_{out}(\theta)$. The phase difference $\phi(\theta)$ between the superconductors is derived from the magnetic flux threading the normal region, governed by the relation $d\phi/d\theta \propto (r_{out}^2 - r_{in}^2)$.
2. Conventional Case: For conventional junctions, the authors assume a standard current-phase relation $I \propto \sin(\phi)$. They maximize the total supercurrent $I_c = I_0 \max_{\phi_0} \int_0^{2\pi} d\theta \sin[\phi(\theta) + \phi_0]$. To understand non-circular shapes (e.g., squares), they utilize a simplified analytical model where the phase evolution includes a perturbation term dependent on the number of corners ($n_c$): $\phi(\theta) = n_v\theta + a \sin(n_c\theta)$.
3. Topological Case: For junctions on 3DTI surfaces, the low-energy physics is modeled using counter-propagating chiral Majorana modes. The effective Hamiltonian couples these modes via a term proportional to $\cos(\phi(\theta)/2)$.
4. Numerical Implementation: To handle the topological case, the authors discretize the Hamiltonian using a variant of the Grover–Sheng–Vishwanath model (a two-leg ladder of Majorana fermions). They address the Nielsen–Ninomiya theorem and boundary condition complications (periodic vs. anti-periodic depending on vortex number $n_v$) by employing two coupled ladders. The critical current is calculated by diagonalizing the Hamiltonian to find eigen-energies $E_\ell$ and computing the Josephson current $I(\phi_0) = -4e/\hbar \sum_\ell \tanh(E_\ell/2k_BT) (dE_\ell/d\phi_0)$.

Key Results
The study reveals a distinct dependence of superconductivity reentrance on the junction's geometry and topological nature:

• Circular Junctions: Both conventional and topological circular Corbino junctions behave identically. Superconductivity is completely destroyed for any non-zero integer number of vortices ($n_v > 0$), as the phase gradient remains constant, leading to total cancellation of the supercurrent.
• Non-Circular (Conventional) Junctions: In junctions with corners (e.g., a square with $n_c=4$), superconductivity exhibits reentrant behavior. The critical current is non-zero only when the number of vortices $n_v$ is an integer multiple of the number of corners ($n_v = m \cdot n_c$). For a square junction, $I_c \neq 0$ only when $n_v$ is a multiple of 4.
• Non-Circular (Topological) Junctions: In the topological case, the reentrance condition changes. The critical current is non-zero when $n_v = m \cdot (n_c/2)$. For a square junction ($n_c=4$), $I_c$ is non-zero for all even values of $n_v$ (multiples of 2). This represents a period halving compared to the conventional case.
• Odd vs. Even Corners: The period halving effect is only observed when the number of corners $n_c$ is even. For odd $n_c$, the topological and conventional junctions behave qualitatively similarly.
• Josephson Diode Effect: The authors note that breaking inversion symmetry in the topological Corbino junction leads to a Josephson diode effect, with polarity switching between even and odd $n_v$.

Significance and Claims
The paper posits that Corbino Josephson junctions serve as experimentally accessible probes of the current-phase relation. The primary significance of this work lies in the identification of period halving in the reentrant superconductivity of non-circular topological junctions as a potential marker for non-trivial topology.

The authors claim that this geometric dependence offers a clearer distinction between topological and trivial scenarios than previous planar junction studies, where deviations from zero were subtle. They suggest that observing the $n_v = n_c/2$ reentrance (specifically the first half-harmonic) could help establish the existence of topological superconductivity in hybrid topological insulator–superconductor junctions.

Limitations and Caveats
The authors explicitly acknowledge several limitations:

• The model assumes the 3DTI bulk gap is large enough to prevent mixing with bulk states.
• It assumes the magnetic flux penetrates only the normal region, neglecting flux trapping in thin superconductors.
• The period halving effect is most visible in the first half-harmonic ($n_v = n_c/2$) because the critical current envelope decays roughly as $1/n_v$.
• While the period halving is consistent with a $\sin(2\phi)$ term in the CPR, the authors caution that alternative mechanisms could theoretically lead to similar effects, though none were identified in their analysis.
• Experimental complications, such as disorder induced by contacts to the inner and outer superconductors, are not included in the model.