🔬 mesoscale physics

Topological and kinetic origins of fractional thermal conductance at the topological insulator--superconductor interface

This paper establishes a theorem proving that thermal conductance alone cannot distinguish between topological and kinetic origins of fractional thermal transport at topological insulator–superconductor interfaces, demonstrating that the observed half-integer plateau in recent experiments arises from full equilibration of co-propagating modes rather than non-Abelian topology, and proposing that charge-sector measurements like anomalous Lorenz ratios and shot noise are required to resolve this ambiguity.

Original authors: Laurence A. Jacobs, Guillermo R. Zemba

Published 2026-10-08

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CC BY 4.0

Original authors: Laurence A. Jacobs, Guillermo R. Zemba

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the strange world of quantum materials, scientists have long searched for a specific kind of order that behaves differently from the familiar laws of physics. Imagine a material where electricity flows without resistance, but only along its edges, like a river confined to a narrow channel. In some of these exotic states, the particles moving along the edge are not just electrons, but something more fundamental and elusive, known as Majorana modes. These particles are their own antiparticles and are believed to hold the key to building powerful quantum computers. For years, the standard way to prove these modes exist was to measure how well the material conducts heat. The theory predicted that if these special particles were present, the heat flow would settle at a precise, half-integer value, a number that seemed impossible to achieve by any other means. This half-integer signal was treated as the smoking gun for a new kind of topological order.

However, a recent experiment using a special type of graphene created a puzzle. Researchers observed this exact same half-integer heat signal, but they knew it was not caused by the exotic Majorana particles. Instead, it was the result of ordinary electrons and holes mixing and balancing their temperatures as they traveled together. This discovery threatened to confuse the field: if a mundane process could mimic the signature of a topological miracle, how could scientists ever be sure what they were seeing? The question was no longer just about finding the particles, but about distinguishing between a genuine topological state and a simple thermal accident.

A new theoretical paper by Laurence Jacobs and Guillermo Zemba addresses this confusion by proving a fundamental rule about how heat moves through these quantum networks. The authors demonstrate that once the different types of particles in a system have fully mixed and reached a state of thermal balance, the total amount of heat they carry depends only on a single, unchangeable number associated with the system's geometry. This number is tied to a deep property of the universe called a gravitational anomaly, which acts like a fixed anchor for the heat flow. The paper proves that no matter how complex the microscopic details are, or whether the heat is carried by exotic particles or ordinary ones, if the system is fully balanced, the heat conductance will always be the same for any setup that shares this same anchor number.

This finding leads to a stark conclusion: measuring heat flow alone can never tell the difference between a topological state and a kinetic one if they happen to produce the same number. If two different physical setups present the same data to a heat meter, the meter will give the same reading, and no amount of precision can separate them. The authors show that the coincidence observed in the graphene experiment, where a simple mixing of ordinary particles produced the same number as a Majorana mode, is an arithmetic accident that cannot be undone by thermometry. The only way to solve the mystery is to stop looking at heat and start looking at electric charge.

The paper proposes a new way to test these systems using a specific setup involving a topological insulator, a superconductor, and a region of quantum Hall effect material. In this arrangement, a magnetic field creates tiny whirlpools, or vortices, in the superconductor. These vortices carry a fraction of an electric charge, specifically one-quarter of the charge of a single electron. The boundary of this system hosts the chiral Majorana mode, which carries heat but no electric charge. The researchers argue that while the heat flow might look identical whether the system is topological or just a result of mixing, the electric behavior will be completely different.

In the topological scenario, the heat is carried by the neutral Majorana mode, while the electric charge is carried by the fractional vortices. This creates a mismatch: the material conducts heat well but conducts electricity poorly, breaking a standard rule of physics known as the Wiedemann-Franz law. In contrast, if the heat signal comes from the kinetic mixing of ordinary charged particles, both the heat and the electricity will follow the standard rules, maintaining a fixed ratio between them. Furthermore, the electric noise generated by the system will behave differently. If the signal is topological, the noise will grow as the magnetic field increases, because more vortices are created to carry the fractional charge. If the signal is kinetic, the noise will stay locked to specific plateaus, ignoring the magnetic field changes.

The authors emphasize that this distinction is not just a suggestion but a necessary consequence of their theorem. They point out that if the experimental setup is not carefully designed, the electric charge carriers can be converted into neutral particles right at the contact point, effectively erasing the very signature needed to solve the puzzle. This would leave the heat meter blind to the difference, exactly as their theorem predicts. To avoid this, the setup must keep the charged and neutral paths separate until they reach the measurement point.

Ultimately, the paper reframes the entire search for these exotic states. It establishes that the question of whether a fractional heat signal is topological or kinetic cannot be answered by heat measurements alone. The answer lies entirely in the electric sector, in the way charge and noise behave. The researchers provide a clear roadmap for future experiments: measure the heat to see the number, but measure the electricity and the noise to understand the mechanism. This approach turns a potential dead end into a precise diagnostic tool, ensuring that when scientists finally find the Majorana modes, they will know it for certain, not by a lucky number, but by the unique fingerprint of the charge they carry.

Technical Summary: Topological and Kinetic Origins of Fractional Thermal Conductance at the Topological Insulator–Superconductor Interface

Problem Statement
Recent experiments, notably by Roy et al. [Nat. Commun. 17, 2853 (2026)], demonstrated a robust half-integer thermal conductance plateau ( $\frac{1}{2}\kappa_0 T$ ) in a bilayer graphene bipolar junction. While such plateaus are conventionally interpreted as direct evidence of non-Abelian topological order (specifically chiral Majorana edge modes), Roy et al. showed that this specific result arises from the full thermal equilibration of co-propagating electron and hole edge modes—a kinetic phenomenon rather than a topological one. This creates a fundamental ambiguity: can thermal conductance ( $\kappa_{th}$ ) measurements distinguish between a genuine topological origin (anomaly-pinned) and a kinetic origin (equilibration-driven) when both yield identical fractional values? The paper addresses whether $\kappa_{th}$ is a sufficient probe to resolve the "topology-versus-kinetics" question.

Methodology
The authors employ a theoretical framework based on boundary conformal field theory (CFT) and thermal transport kinetics.

Theorem Formulation: They derive a general theorem regarding two-terminal thermal transport in networks of chiral edge segments connected by ideal floating contacts. The derivation relies on the universal energy current of chiral edge CFTs and the assumption of full thermal equilibration within segments.
Network Analysis: Using heat-balance calculus, they analyze the steady-state energy flow through networks, treating floating contacts as nodes that absorb incident flux and re-emit it at a single contact temperature.
Application to TI–SC Interfaces: They apply these general results to a specific physical platform: the interface between a three-dimensional time-reversal-invariant topological insulator (TI) and an $s$ $s$ -wave superconductor (SC), coupled laterally to an integer quantum Hall (IQH) region. They examine two distinct realizations of the same boundary anomaly:
- Topological Realization: An isolated chiral Majorana boundary mode arising from the TI–SC interface in a magnetic field.
- Kinetic Realization: A line junction where the TI–SC boundary co-propagates with an adjacent IQH edge, leading to thermal equilibration.
Charge-Sector Diagnostics: Recognizing the limitations of thermal transport, they analyze electrical conductance (via the Lorenz ratio) and excess shot noise as alternative observables that are not constrained by gravitational anomalies.

Key Contributions

Anomaly Pinning Theorem: The paper proves that under full thermal equilibration, the two-terminal thermal conductance of a network of chiral edge segments joined at ideal floating contacts is a rational function of the net chiral central charges ( $c_i$ ) of the segments alone. Crucially, $\kappa_{th}$ depends only on these anomaly coefficients and the network connectivity, remaining invariant under arbitrary local boundary kinetics, disorder, or microscopic details.
Impossibility Corollary: A direct consequence of the theorem is an impossibility statement: whenever two realizations present the same anomaly data to the same network, no thermal-conductance measurement in that equilibrated geometry can distinguish them, regardless of their microscopic origin. If a kinetic equilibration process yields the same central charge as a topological mode (e.g., $c=1/2$ ) within the same network configuration, $\kappa_{th}$ cannot separate them. The theorem is explicitly conditional for composite contacts (e.g., where a Majorana branch and a charged IQH branch meet) if the hypothesis of a single contact temperature fails.
TI–SC–IQH Platform Analysis: The authors develop the TI–SC interface as a testbed for this theorem. They demonstrate that:
- A topological chiral Majorana mode carries $c_M = 1/2$ .
- A kinetic chain of a TI–SC boundary flanking an IQH region (TI–SC | IQH | TI–SC) yields an equilibrated conductance of $\kappa_{th} = \frac{|\nu_{IQH}|}{1 + 4|\nu_{IQH}|} \kappa_0 T$ .
- While these values differ for most fillings, the theorem establishes that if they were to coincide (as in the specific $(2, -1)$ graphene case), thermometry would be fundamentally blind to the mechanism.
Charge-Sector Resolution: The paper identifies that the distinction must be sought in the charge sector, which is not "anomaly-pinned" at a superconducting boundary due to charge non-conservation (condensate breaking).
- Lorenz Ratio: The topological response is anomalous ( $L \to \infty$ as $G \to 0$ ), whereas the kinetic response obeys the Wiedemann–Franz law ( $L = L_0$ ). In the composite device, the measured ratio is a specific fraction of $L_0$ determined by the series geometry, distinct from the topological limit.
- Shot Noise: The topological mechanism generates excess noise scaling with the magnetic field as $B^{1/2}$ (due to $e/4$ vortex tunneling), while the kinetic mechanism is piecewise constant between IQH plateaus.

Results

Theorem Validity: The thermal conductance of equilibrated networks is strictly determined by the connectivity and the net chiral central charges ( $c_i$ ). Microscopic details (e.g., whether $c=1/2$ comes from a Majorana mode or equilibrated integer edges) are erased in the thermal sector, provided the network consists of simple segments or composite segments that thermalize to a single temperature.
Device Blindness: In the TI–SC–IQH platform, if the quantum Hall edge is in direct contact with the superconductor (Case C), Andreev conversion transforms charged modes into Majorana modes. This preserves the net central charge (and thus $\kappa_{th}$ ) but destroys the charge sector, rendering the device incapable of distinguishing the mechanism via electrical or noise measurements. The presence of a spacer (Case A) is essential to preserve the charged modes for diagnostics.
Diagnostic Signatures: The paper provides explicit formulas for the distinguishing signatures:
- Lorenz Ratio: $L = L_0 / (1 + 4|\nu_{IQH}|)$ for the kinetic chain, contrasting with the divergent ratio of the isolated topological boundary.
- Shot Noise: The topological signal scales as $S \propto N_v^{1/2} (T_M - T_0)$ , where $N_v \propto B$ , while the kinetic signal is constant between plateaus.

Significance and Claims
The paper claims to reframe the topology-versus-kinetics question by establishing a fundamental limit on thermal transport measurements. It asserts that the coincidence of fractional thermal conductance values (e.g., $\frac{1}{2}\kappa_0 T$ ) arising from different mechanisms is an "arithmetic fact" that thermometry cannot undo when the same anomaly data is presented to the same network. The significance lies in proving that the search for a mechanism-sensitive observable must leave the thermal sector entirely and move to the charge sector (electrical conductance and shot noise).

The authors are modest about the experimental feasibility of the specific composite device, noting that the existence of the required "composite segment" (a co-propagating Majorana branch and a charged IQH branch separated by a spacer that prevents Andreev conversion but allows equilibration) is a non-trivial open problem. They do not claim to have solved the microscopic kinetic theory of the composite contact or the vortex Fano factor, identifying these as necessary future work to complete the theory. The paper's primary contribution is the rigorous theorem of "anomaly pinning" and the identification of the unique charge-sector observables required to resolve the ambiguity it creates.

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