🔬 condensed matter

Parity Anomaly of Preformed Pairs Governs the Thermal Hall Effect above $T_c$

This paper resolves the long-standing puzzle of the large negative thermal Hall effect in the cuprate pseudogap phase by demonstrating that the parity anomaly of preformed pairs yields a parameter-free formula linking the thermal Hall conductivity directly to the experimentally measurable pseudogap energy scale.

Original authors: Kumar Ghosh

Published 2026-07-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Kumar Ghosh

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

Imagine a bustling city where the citizens are electrons. Usually, when these electrons decide to dance in perfect unison (a state called superconductivity), they create a special, invisible highway that lets heat flow in a very specific, one-way direction. This is the "Thermal Hall Effect," and it's a bit like a thermal traffic jam that only moves left or right, depending on the city's layout.

For years, scientists have been puzzled by a strange phenomenon in a family of materials called cuprates. They found that even when the superconducting dance floor was empty—when the electrons had stopped dancing in perfect sync and the "condensate" had vanished—a massive, negative thermal traffic jam still existed. It was as if the highway was still there, even though the parade had ended.

Previous theories tried to explain this ghost highway using ideas like "chiral phonons" (vibrating atoms), "spinons" (magnetic excitations), or "loop currents" (tiny electrical loops). But these theories were like trying to solve a puzzle with missing pieces; they required scientists to guess at "free parameters" (magic numbers they could tweak to make the math work) and couldn't explain why the signal appeared exactly when it did.

The Big Discovery: The "Ghost" Highway is Real

This paper proposes a new, crystal-clear explanation that needs no magic numbers at all. The authors suggest that the secret lies in something called the parity anomaly, a deep rule from quantum physics that acts like a hidden law of the universe.

Think of the electrons in these materials as pairs of dancers. Even when the big, synchronized parade (superconductivity) stops, the pairs don't immediately break up. They linger, forming what the paper calls "preformed pairs." These pairs are like couples holding hands in a crowd, even though they aren't marching in a line yet.

The paper argues that these lingering pairs create a "mass gap"—a kind of energy barrier that acts exactly like the mass of a particle. In the world of quantum field theory, having this mass is enough to trigger the parity anomaly. It's as if the mere presence of these hand-holding couples is enough to twist the fabric of space-time just enough to create that one-way thermal highway, even without the full parade.

The Magic Formula

The authors derived a precise, parameter-free formula to describe this effect. It looks like this:

$\kappa_{xy}/T = (\pi^2 k_B^2 / 6h) \times C \times \tanh[\Delta_{pg}(T) / (2k_B T)]$

Let's break down the fun parts:

$\kappa_{xy}/T$ : This is the strength of the thermal traffic jam.
$C$ : This is a "Chern number," a topological integer that acts like a fingerprint of the material's shape. For the cuprates discussed, this is exactly 1.
$\Delta_{pg}(T)$ : This is the "pseudogap," the energy scale of those lingering preformed pairs. It's a number you can actually measure in a lab using tools like ARPES or STM.
$\tanh$ : This is a mathematical function that acts like a dimmer switch. When the temperature is high, the switch is off. As the temperature drops and the pairs get stronger, the switch turns on, and the thermal signal grows.

The most exciting part? This formula predicts that the thermal signal should turn on at a specific temperature called $T^*$ (the pseudogap onset), which is much higher than the temperature where superconductivity usually starts ( $T_c$ ). It's like the traffic jam starts forming the moment the couples start holding hands, long before they start marching in a parade.

What This Paper Rules Out

The authors are very clear about what this is not. They explicitly argue against the idea that you need a fully formed, coherent superconducting condensate to see this effect. If you think the signal only appears because of a perfect, long-range dance, this paper says you're wrong. The signal exists even in a Mott insulator (a material with no superconductivity at all), like undoped La $_2$ CuO $_4$ , where the signal is actually huge.

They also rule out the competing theories mentioned earlier. The paper states that proposals based on chiral phonons, spinons, or loop currents fail because they cannot predict the magnitude and temperature dependence without using those "free parameters" (guesswork). This new theory needs no guesswork; once you measure the gap ( $\Delta_{pg}$ ), the whole story is fixed.

How Sure Are They?

The authors are incredibly confident, but they back it up with rigorous checks rather than just hand-waving.

Mathematical Proof: They rely on a "Coleman-Hill non-renormalization" theorem. This is a fancy way of saying that once you have the basic ingredients, the universe doesn't allow for messy, higher-order corrections to mess up the result. The math is "exact."
Computer Simulations: They didn't just stop at the math. They ran two different types of massive computer simulations:
- Wilson-loop flux threading: They simulated cylinders of the material and threaded magnetic flux through them. The result? The "winding number" (a count of how many times the quantum state twists) was exactly 1.000000 across all sizes tested.
- DMRG (Density Matrix Renormalization Group): They simulated the many-body physics of the system. They found that the "edge currents" (the flow of heat at the boundary) matched their predictions with 0.2% accuracy.
- Crucially, they checked the size of the system. If the effect were due to some other, simpler reason, the signal would change in a specific way as the system got bigger (a "power-law" change). Instead, they found the changes were purely exponential, exactly as their "parity anomaly" theory predicts. This is a very strong signature that the mechanism is real.

What's Next?

The paper doesn't claim to have solved the mystery of high-temperature superconductivity entirely, but it has solved a very specific, long-standing puzzle: the origin of the thermal Hall signal above $T_c$ .

They offer a "falsifiable" test for other scientists. If you measure the thermal Hall signal in a cuprate or a "magic-angle twisted bilayer graphene" (MATBG) sample, you should see the signal turn on at $T^*$ , not $T_c$ . Furthermore, if you take the "logarithmic derivative" (a specific way of looking at how the signal changes with temperature), it should perfectly track the independently measured pseudogap size.

For MATBG, they predict a specific, saturated value for the thermal Hall conductance of roughly $4.73 \times 10^{-13}$ W K $^{-2}$ (assuming a Chern number of 1). If experiments see this number, and if the signal disappears in samples where the pseudogap is quenched, the theory wins. If not, the theory is busted.

In short, this paper suggests that the "ghost" thermal highway isn't a ghost at all. It's a real, robust feature of nature, created by the simple act of electron pairs holding hands, governed by a deep, unbreakable law of quantum geometry.

Technical Summary: Parity Anomaly of Preformed Pairs Governs the Thermal Hall Effect above $T_c$

Problem Statement
Recent experiments have reported a large negative thermal Hall signal ( $\kappa_{xy}/T$ ) in the pseudogap phase of multiple cuprate families (Nd-LSCO, Eu-LSCO, LSCO, Bi2201) and even in the undoped Mott insulator La $_2$ CuO $_4$ . This signal persists in the field-induced normal state where the superconducting condensate order parameter ( $\Delta_{sc}$ ) has vanished ( $T > T_c$ ). Existing microscopic theories, including those based on chiral phonons, spinons, or loop currents, fail to reproduce the magnitude and temperature dependence of this signal without introducing free parameters or undetermined coupling constants. Crucially, no existing theory predicts the signal's onset at the pseudogap temperature $T^*$ rather than the superconducting transition $T_c$ , nor does it link the signal to an independently measured spectroscopic gap.

Methodology and Theoretical Framework
The paper proposes that the observed thermal Hall effect is governed by the parity anomaly of (2+1)-dimensional quantum field theory, specifically applied to preformed chiral pairs in the pseudogap regime. The derivation relies on three theoretical pillars:

Exact Finite-Temperature Chern-Simons Level: Using the framework of a massive Dirac fermion on a Euclidean cylinder ( $\mathbb{R} \times \mathbb{R} \times S^1$ ), the authors utilize exact holonomy-resummed kernels derived via Pauli-Villars regularization and Ginsparg-Wilson lattice methods. This establishes the parity-odd effective action without gradient expansions.
BCS-BEC Crossover and Two-Gap Structure: The authors adopt the BCS-BEC crossover formalism to decompose the total fermionic excitation gap $\Delta$ into a condensate gap $\Delta_{sc}$ and a pseudogap $\Delta_{pg}$ (from non-condensed preformed pairs). Above $T_c$ , $\Delta_{sc} = 0$ , but $\Delta_{pg} \neq 0$ .
Coleman-Hill Non-Renormalization: The authors invoke the Coleman-Hill theorem, which guarantees that the topological mass term (Chern-Simons level) in (2+1)D QED receives radiative corrections only at the one-loop level. This ensures that once the gap $\Delta_{pg}(T)$ is determined, the relationship to the thermal Hall response is exact and parameter-free.

Key Contributions and Results
The central result is an exact, parameter-free formula for the thermal Hall conductance in the pseudogap window ( $T_c < T < T^*$ ):
$\frac{\kappa_{xy}}{T} = \frac{\pi^2 k_B^2}{6h} C \tanh\left[\frac{\Delta_{pg}(T)}{2k_B T}\right]$
where $C$ is the Chern number of the chiral pairing channel (determined by band topology) and $\Delta_{pg}(T)$ is the pseudogap magnitude measurable via ARPES or STM.

Parity Anomaly as the Mechanism: The paper demonstrates that the preformed-pair gap $\Delta_{pg}(T)$ enters the parity-odd fermion determinant identically to a condensate mass. Consequently, the system supports a non-trivial finite-temperature parity-anomaly response even in the absence of long-range phase coherence.
Topological Robustness: The Chern number $C$ remains well-defined above $T_c$ as long as $\Delta_{pg}(T) > 0$ . For weak-pairing regimes relevant to cuprates and magic-angle twisted bilayer graphene (MATBG), $C = +1$ .
Finite-Size Scaling ( $c_1=0$ Theorem): The paper proves that finite-size corrections to the thermal Hall signal on a spatial cylinder are purely exponential ( $\sim e^{-L/\xi}$ ) with no power-law ( $1/L^n$ ) terms. This is a direct consequence of the $c_1=0$ theorem for the Poisson kernel in the presence of a bulk gap.
Numerical Validation:
- Wilson-Loop Flux Threading: Calculations on BdG cylinders ( $L_y = 4$ to $14$) confirm that the flux-threaded response exhibits exactly quantized cycle winding ( $\Delta P = 1$ ) and that local deviations decay exponentially with cylinder width, consistent with the theoretical correlation length $\xi_{pg} = v_F/\Delta_{pg}$ .
- DMRG Simulations: Density Matrix Renormalization Group (DMRG) simulations on $p+ip$ cylinders confirm the anomaly-controlled finite-size structure. The chiral edge currents satisfy $J_{left} = -J_{right}$ to machine precision, and the extracted correlation lengths match theoretical predictions to within 0.2% accuracy. Entanglement entropy analysis confirms the absence of power-law corrections, validating the $c_1=0$ theorem at the many-body level.

Experimental Predictions and Significance
The paper claims to resolve the long-standing puzzle of the pseudogap thermal Hall effect by providing the first parameter-free prediction that links the signal directly to the independently measured spectroscopic gap $\Delta_{pg}(T)$ .

Falsifiable Test: The theory predicts that the thermal Hall signal should onset at the pseudogap scale $T^*$ , not $T_c$ . A critical experimental test involves measuring the logarithmic derivative of the signal:
$\frac{d}{dT} \ln \left| \frac{\kappa_{xy}}{T} \right| \stackrel{?}{=} \frac{d}{dT} \ln \tanh\left[\frac{\Delta_{pg}(T)}{2k_B T}\right]$
Success requires the signal to track the temperature dependence of $\Delta_{pg}(T)$ through the $\tanh$ factor.
Cuprates: The theory explains the large negative signal in underdoped cuprates and the signal in undoped La $_2$ CuO $_4$ (a Mott insulator) as a consequence of the parity anomaly of preformed pairs, independent of superconducting phase coherence.
MATBG: The paper provides a quantitative target for magic-angle twisted bilayer graphene. Using tunneling data where a pseudogap $\Delta_T$ persists above $T_c$ , the theory predicts a saturated thermal Hall conductance of $\kappa_{xy}/T \approx \pi^2 k_B^2 / 6h$ (for $C=1$ ) just above $T_c$ . It further predicts that this signal should vanish in hBN-aligned devices where the pseudogap is quenched.

Conclusion
The paper establishes a direct, non-perturbative link between (2+1)-dimensional parity anomaly physics and thermal transport in strongly correlated materials. It argues that the condensate is not required for a quantized thermal Hall response; rather, the existence of a fermionic mass gap (the pseudogap) is sufficient to generate the anomaly. This framework offers a unified explanation for thermal Hall transport in chiral pseudogap phases, distinguishing itself from competing proposals by its lack of free parameters and its specific, testable dependence on the spectroscopic gap.

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