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Gravitational anomaly spectroscopy of a Kitaev spin liquid

This paper proposes and validates a spectroscopic method to measure the chiral central charge $c$ and conformal spin of Kitaev spin liquids by analyzing edge absorption weights governed by the Virasoro algebra, demonstrating its robustness against thermal states, disorder, and finite-size effects across classical and quantum simulations.

Original authors: Kumar Ghosh

Published 2026-10-30

📖 5 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

In the strange world of quantum materials, some substances behave less like ordinary solids and more like a single, giant quantum object. Among these, a special class known as topological phases holds a secret in their very structure. Unlike a standard magnet where the magnetic properties come from the alignment of tiny atomic spins, these materials derive their behavior from the global shape of their quantum state. A key feature of certain two-dimensional topological phases is that they host special waves that can only travel in one direction along the material's edge, much like a one-way street for energy. The strength of this one-way flow is determined by a single number, a fundamental constant of the material that physicists call the chiral central charge. This number is not just a label; it dictates how much heat the edge can carry and is deeply tied to a subtle imbalance in how the material responds to gravity, a phenomenon known as a gravitational anomaly. Measuring this number has been notoriously difficult. Previous attempts relied on measuring heat flow, but heat is messy and easily confused by vibrations in the material's atoms, or they required the material to be in a perfectly pure, frozen state, which is nearly impossible to maintain in a real experiment.

A researcher has now proposed and tested a new way to find this elusive number, one that works even when the material is warm or jiggling, and does not require a perfect starting state. Instead of trying to measure heat, they looked at how the edge of the material absorbs sound-like waves of energy. Imagine sending a ripple along the edge of a pond; the researcher showed that if you send a traveling wave that matches the rhythm of the edge's natural vibrations, the amount of energy the edge soaks up reveals the hidden number they are looking for. By comparing how much energy is absorbed by a traveling wave against how the edge responds to a simple, static push, they found a mathematical recipe that cancels out all the messy, state-dependent details. This recipe leaves behind only the fundamental number, the chiral central charge, regardless of whether the material is cold, hot, or somewhere in between.

The researcher tested this idea using a specific model of a quantum spin liquid, a theoretical material made of spins arranged in a honeycomb pattern. Using powerful computer simulations, they modeled systems with up to nearly 7,200 spins to see if their method held up. They simulated sending traveling waves along the edge of these virtual materials and measured the absorption. The results were strikingly precise. In the largest simulations, the method recovered the theoretical value of the central charge to within five decimal places. Even more importantly for real-world applications, the method worked with high accuracy on much smaller systems, requiring only about 100 spins to get the answer within one percent of the true value. This suggests that the technique is robust enough to be used on current and next-generation quantum computers, which can only handle a few hundred quantum bits at a time.

The study also explored how well this method survives the imperfections of the real world. They tested the approach on materials that were not in a perfect ground state but were instead at a finite temperature, or "kicked" by external pulses to create non-stationary states. The method remained accurate, provided the material was not so hot that the edge waves were drowned out by the bulk of the material. They also checked what happens if the material contains random defects or if the connections between the spins are slightly off-target. The results showed that the measurement is remarkably resilient to these errors, staying accurate even when the connections between spins vary by up to twenty percent. However, the method does have a clear limit: it fails if there are unwanted particles floating in the middle of the material, known as bulk quasiparticles. This sets a specific requirement for any future experiment: the interior of the material must be kept clear of these excitations.

Beyond just finding the central charge, the researcher showed that their technique could also reveal another hidden property of the material: the "spin" of exotic particles called anyons that live inside the bulk. These anyons are the building blocks for a proposed type of quantum computer that is naturally protected from errors. By comparing the absorption signals when an anyon is present versus when it is absent, the researcher could calculate the specific type of anyon without needing any external calibration. In their simulations, this ratio matched the theoretical prediction perfectly. This dual ability to measure both the edge's fundamental charge and the bulk's particle properties makes the technique a powerful diagnostic tool.

The researcher then took their idea a step further by simulating how a real quantum computer, specifically one using trapped ions, would carry out this measurement. They modeled the entire process, including the specific sequence of gates used to create the material and the pulses used to probe it. Their simulation showed that even with the imperfections and noise inherent in current quantum hardware, the method could still extract the correct central charge with an error of only about three and a half percent. This suggests that the experiment is within reach of near-term technology. The study concludes that by listening to the edge of a quantum material with a traveling wave, scientists can finally hear the fundamental number that defines its topological nature, a measurement that was previously thought to be out of reach for anything other than a perfect, frozen state. This opens a new path for verifying that quantum simulators are truly creating the exotic states of matter they are designed to produce.

Technical Summary: Gravitational Anomaly Spectroscopy of a Kitaev Spin Liquid

Problem Statement
The chiral central charge, $c$ , is a fundamental invariant of two-dimensional topological phases with broken time-reversal symmetry. It quantifies the gravitational anomaly and fixes the thermal Hall conductance of the chiral edge modes. However, measuring $c$ experimentally has proven difficult. Existing thermal transport methods are complicated by phonon contributions, and wave-function probes (such as the modular commutator) require a pure ground state and full tomography, which are challenging to achieve in current quantum processors. Furthermore, in one-dimensional critical chains, $c$ often appears only as an overall energy shift that is difficult to resolve, and in non-chiral systems, the gravitational anomaly vanishes. The central question addressed is whether $c$ can be extracted from a spectroscopic measurement on a quantum processor without requiring a pure ground state.

Methodology
The author proposes a spectroscopic protocol based on the Virasoro algebra of the edge conformal field theory (CFT). The core idea is that a traveling modulation of the edge couplings couples to the edge stress tensor, $T(x)$ .

Traveling Drive: A weak modulation of edge bond couplings, $\lambda O_n$ , travels along the edge with wavevector $Q_n = 2\pi n/L$ . This drive couples to the Fourier modes of the stress tensor, $L_{-n}$ .
Absorption Weight: The absorbed weight, $W_n$ , for a drive at harmonic $n$ is proportional to the expectation value of the Virasoro commutator, $\langle [L_n, L_{-n}] \rangle$ . According to the Virasoro algebra, this yields:
$W_n \propto 2n\langle L_0 \rangle + \frac{c}{12}(n^3 - n)$
Here, $\langle L_0 \rangle$ is a state-dependent term (related to the energy of the edge state), while $c$ is the universal central charge.
Calibration: To eliminate the unknown coupling prefactor and the state-dependent $\langle L_0 \rangle$ , the author proposes a calibration step. A static change of the same couplings shifts the edge energy levels. The rate of this shift, $\partial \omega_F / \partial \lambda$ , provides a normalization factor.
Extracting Observables: By defining a calibrated weight $D_n = W_n / (\partial \omega_F / \partial \lambda)^2$ $D_{n} = W_{n} / (\partial ω_{F} / \partial λ)^{2}$ , the state dependence can be isolated.
- Central Charge ( $c$ ): A linear combination of the first two harmonics cancels the state-dependent term: $C \equiv 2D_2 - 4D_1 = c$ . This holds for any edge state (thermal, mixed, or non-stationary).
- Conformal Spin ( $h_\sigma$ ): The ratio of line weights between the sector with a bulk vison (Ramond sector) and without (Neveu-Schwarz sector), $\Lambda = W_1^R / W_2^{NS}$ , yields $4h_\sigma/c$ . This ratio requires no calibration of the drive amplitude.

Key Results
The author validates this protocol using exact diagonalization and Density Matrix Renormalization Group (DMRG) simulations on the Kitaev honeycomb model (where the theoretical value is $c=1/2$ and $h_\sigma=1/16$ ).

Accuracy: On large systems (up to 7168 spins), the method recovers $c = 0.49999$ and $h_\sigma = 0.06250$ .
Finite Size: With the "line calibration" (using the shift of the measured absorption line itself), accurate results ( $\sim 1\%$ error) are achievable on small systems of 96 spins.
Robustness:
- State Independence: The value of $C$ remains within 1% of the true $c$ even in thermal edge states and non-stationary states (created by "kicks"), provided the temperature is below a threshold determined by the edge dispersion curvature.
- Disorder: The method is robust against random coupling errors (up to 20%) and static vison disorder, provided visons are kept at least five lattice spacings away from the edge.
- Interactions: When edge Majoranas interact (simulated via DMRG), the calibration absorbs the renormalization of the edge velocity and drive strength, keeping $C$ within 1.5% of the ideal value even as the edge velocity changes by $\pm 13\%$ .
Failure Conditions: The protocol fails if bulk Majorana quasiparticles are present, as bulk transitions can fall into the edge spectral window. It also requires pulses to be slow relative to the bulk gap to avoid exciting bulk modes.
Processor Protocol: The author simulates a complete protocol for a quantum processor, including a native Floquet circuit (mimicking trapped-ion hardware) and echo readout. They estimate that 96 to 600 spins with $10^5$ to $10^6$ shots per setting are sufficient to measure $c$ and $h_\sigma$ with high precision.

Significance and Claims
The paper claims to demonstrate that the chiral central charge is a spectroscopic quantity accessible via edge absorption measurements, distinct from thermal transport or ground-state tomography.

State Independence: Unlike previous methods, this approach does not require a pure ground state; it functions in thermal and non-stationary states naturally produced by processors or materials.
Edge Isolation: By utilizing the chirality and frequency of the drive, the method isolates the edge response from bulk contributions (including phonons in materials), avoiding the need for heat flow equilibration.
Topological Characterization: The protocol simultaneously yields the conformal spin of the non-Abelian Ising anyon ( $h_\sigma$ ) via the calibration-free ratio $\Lambda$ , providing a direct test of the topological order.
Generality: The sum rule relies only on the Virasoro algebra, suggesting applicability to other chiral edges (e.g., quantum Hall edges, chiral superconductors) provided a drive couples to the stress tensor.

The author concludes that this method offers a direct, resource-efficient test for the presence of chiral topological order in quantum simulations, with a clear set of experimental requirements (bulk vacuum, slow pulses) and a demonstrated tolerance to realistic noise and interactions.

Technical Summary: Gravitational Anomaly Spectroscopy of a Kitaev Spin Liquid

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