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arXiv:2610.38704

Gravitational anomaly spectroscopy of a Kitaev spin liquid

Kumar Ghosh

cond-mat.str-elcond-mat.mes-hallquant-phSubmitted Oct 28, 2026• Announced Oct 30, 2026• 0 citations
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Abstract

The chiral central charge $c$ measures the gravitational anomaly of a two-dimensional topological phase. It fixes the heat current carried along the edge. So far, it has been accessed through thermal transport, where phonons complicate the signal, or through wave-function probes that need a pure ground state. Here we show that $c$ is a spectroscopic quantity of the edge. A weak modulation of the edge couplings that travel along the edge is absorbed according to the Virasoro algebra. Normalized by the response to a static change of the same couplings, the absorbed weights obey $D_n=2n\langle L_0\rangle+\frac{c}{12}(n^3-n)$, so that $\mathcal{C}\equiv2D_2-4D_1=c$ in any state of the ideal edge theory, thermal or not. The central charge thus appears in transition weights, although it cancels from excitation gaps. The ratio of line weights with and without a vison in the bulk, $Λ=4h_σ/c$, needs no calibration and fixes the conformal spin $h_σ$ of the non-Abelian Ising anyon. For the Kitaev honeycomb spin liquid we find $\mathcal{C}=0.49999$ and $h_σ=0.06250$ on 7168 spins. Calibrating with the shift of the measured line itself gives $c$ within about $1\%$ on 96 spins and within $1\%$ under $20\%$ random coupling errors. Interacting edge Majoranas in a fixed flux sector, treated with DMRG, keep $\mathcal{C}$ within $1.5\%$ while the edge velocity changes by $\pm13\%$. An exact simulation of a quantum-processor protocol, including the native Floquet circuit, returns $c$ within $3.5\%$ on 96 to 600 spins.

AI Summary

Can the chiral central charge of a two-dimensional topological phase be extracted from edge spectroscopy without thermal transport or pure-state tomography? This paper proposes a traveling modulation of edge couplings that projects onto stress-tensor modes. Normalizing directional absorption weights by the edge-level-spacing response to a static modulation gives $D_n=2n\langle L_0\rangle+c(n^3-n)/12$. Consequently, $2D_2-4D_1=c$ independently of the ideal edge-CFT state. A ground-state line-weight ratio between sectors with and without a bulk vison gives $4h_\sigma/c$, providing access to the Ising anyon’s conformal spin. Exact free-Majorana calculations for the Kitaev honeycomb model yield $c=0.49999$ and $h_\sigma=0.06250$ on 7,168 spins. Calibration using the measured absorption line’s own frequency shift reduces finite-size errors: a 96-spin flake gives $|c|=0.5055$. Tests include thermal and nonstationary edge states, static visons and random couplings; suitable flakes remain within about $1\%$ under $20\%$ coupling errors. Fixed-flux-sector DMRG with four-Majorana interactions finds variation within $1.5\%$ while edge velocities change by approximately $\pm13\%$. Processor readouts and a native Floquet circuit are simulated separately. Echo simulations recover $c$ within $3.5\%$ on 96–600 spins, but assume pure initial states; correlator readout supports mixed states. Proposed measurements require roughly $25/J$ coherent evolution and $10^5$–$10^6$ shots per setting. Limitations include bulk-quasiparticle contamination, dispersion curvature, boundary-mode control and decoherence-induced spectral leakage. Interactions are tested only with fixed gauge flux; the full Kitaev–Heisenberg/field problem and multibranch edges remain open.

No sign of AI-generated textStrong assumptions

Error found: The abstract omits that the calibration-free identity Λ = 4h_σ/c is established for the two sector ground states, rather than arbitrary edge states. The abstract describes the processor-protocol accuracy as including the native Floquet circuit, although the reported echo-protocol and Floquet-Hamiltonian spectroscopy simulations are separate tests.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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Can the chiral central charge of a two-dimensional topological phase be extracted from edge spectroscopy without thermal transport or pure-state tomography? This paper proposes a traveling modulation of edge couplings that projects onto stress-tensor modes. Normalizing directional absorption weights by the edge-level-spacing response to a static modulation gives Dn=2n⟨L0⟩+c(n3−n)/12D_n=2n\langle L_0\rangle+c(n^3-n)/12Dn​=2n⟨L0​⟩+c(n3−n)/12. Consequently, 2D2−4D1=c2D_2-4D_1=c2D2​−4D1​=c independently of the ideal edge-CFT state. A ground-state line-weight ratio between sectors with and without a bulk vison gives 4hσ/c4h_\sigma/c4hσ​/c, providing access to the Ising anyon’s conformal spin.
Exact free-Majorana calculations for the Kitaev honeycomb model yield c=0.49999c=0.49999c=0.49999 and hσ=0.06250h_\sigma=0.06250hσ​=0.06250 on 7,168 spins. Calibration using the measured absorption line’s own frequency shift reduces finite-size errors: a 96-spin flake gives ∣c∣=0.5055|c|=0.5055∣c∣=0.5055. Tests include thermal and nonstationary edge states, static visons and random couplings; suitable flakes remain within about 1%1\%1% under 20%20\%20% coupling errors. Fixed-flux-sector DMRG with four-Majorana interactions finds variation within 1.5%1.5\%1.5% while edge velocities change by approximately ±13%\pm13\%±13%.
Processor readouts and a native Floquet circuit are simulated separately. Echo simulations recover ccc within 3.5%3.5\%3.5% on 96–600 spins, but assume pure initial states; correlator readout supports mixed states. Proposed measurements require roughly 25/J25/J25/J coherent evolution and 10510^5105–10610^6106 shots per setting. Limitations include bulk-quasiparticle contamination, dispersion curvature, boundary-mode control and decoherence-induced spectral leakage. Interactions are tested only with fixed gauge flux; the full Kitaev–Heisenberg/field problem and multibranch edges remain open.
In the Kitaev honeycomb model, a weak magnetic field gaps the gapless Majorana cone, driving the system into a gapped non-Abelian phase with Ising topological order. The edge supports a single chiral Majorana mode with chiral central charge c−=1/2c_- = 1/2c−​=1/2, yielding a half-quantized thermal Hall conductance κxy/T=π2kB2/6h\kappa_{xy}/T = \pi^2 k_B^2/6hκxy​/T=π2kB2​/6h. We rigorously investigate whether observing this half-quantized value uniquely identifies the non-Abelian Ising anyon sector, or whether Abelian topological orders or extrinsic mechanisms can produce the same signature. By combining the edge-bulk correspondence with the classification of strictly two-dimensional bosonic topological orders, we show that a chiral central charge c−=1/2c_-=1/2c−​=1/2 (mod 8) is the unique fingerprint of the Ising non-Abelian anyon theory among minimal bosonic topological orders. Abelian bosonic orders necessarily have integer c−c_-c−​, with the minimal nonzero value being c−=1c_-=1c−​=1. Thus, a robustly half-quantized κxy/T\kappa_{xy}/Tκxy​/T uniquely distinguishes the Majorana sector from Abelian alternatives, provided the system is a genuine 2D bosonic spin system free of invertible E8E_8E8​ stacking or phonon contamination. ## 1. Introduction The Kitaev honeycomb model is an exactly solvable model of a quantum spin liquid, characterized by fractionalized excitations known as anyons. In the absence of a magnetic field, the model hosts a gapless Majorana cone. A weak magnetic field in the [111] direction gaps this cone and induces a non-Abelian phase with Ising topological order. This phase supports a single chiral Majorana edge mode, which contributes to a quantized thermal Hall conductance. The central question addressed in this preprint is whether the observation of a half-quantized thermal Hall conductance, corresponding to a chiral central charge c−=1/2c_- = 1/2c−​=1/2, uniquely identifies the non-Abelian Ising anyon sector, or if Abelian topological orders can mimic this signature. ## 2. Background The thermal Hall conductance $\kappa_{x
Majorana fermions arise in a broad range of systems in condensed matter physics, not only as a useful abstraction but also as emergent quasiparticles, e.g., at the boundaries of topological insulators. In this dissertation, we consider two distinct settings wherein Majorana fermions play a central role: (i) non-abelian Kitaev spin liquids and (ii) entanglement dynamics in monitored quantum circuitsThe Kitaev spin liquid is a paradigmatic model featuring non-abelian anyons and a chiral Majorana edge mode when time-reversal symmetry is broken. To this end, it presents an alluring platform for realizing topologically protected quantum information. However, experimental efforts to identify this putative phase have been hampered both by the difficulty of probing the charge neutral quasiparticles and by the presence of phonons and other deviations from the idealized toy model. Here we examine how judicious device design may suppress bulk phonon contributions to thermal transport, allowing for refined observation of transport mediated by the chiral Majorana edge mode. Moreover, we show that the bulk topological order can be revealed through universal features in the tunneling conductance across a point contact. Looking forward, we develop a roadmap for near-term experimental devices to demonstrate robust control of topologically protected qubits encoded in the bulk anyons.Much as Majorana models form the building blocks for symmetry-protected topological order, Majorana circuits provide a powerful framework for studying the entanglement dynamics and phases of monitored quantum circuits. Here we show that the entanglement dynamics of Majorana circuits can be mapped exactly to the statistical mechanics of classical loop models. Harnessing this framework, we undertake a precise numerical characterization of the universality of entanglement transitions in symmetry classes BDI and D in both (1+1) and (2+1)-dimensional circuits. Furthermore, we employ duality relations for loop models to construct an analytically solvable interacting circuit which features not only the usual percolation criticality but also Berezinskii-Kosterlitz-Thouless critical lines and a tricritical point with Ising universality. Lastly, by generalizing to colored loop models for Majorana circuits with classical control, we provide a blueprint for engineering even more diverse entanglement dynamics which may include Hilbert-space fragmentation and subdiffusive transport.
A quantum spin liquid is a magnetic phase of matter that is remarkable for its ground state long-range entanglement and fractional excitations. A quantum spin liquid appears in the well-known Kitaev honeycomb spin-half model. In a magnetic field, there is a phase transition to a topological phase with an energy gap in the bulk and chiral Majorana fermions at the edge. However, recent nuclear magnetic resonance experiments suggest excitations without a gap in this phase, signalled by the cubic temperature dependence of the spin-lattice relaxation rate at low temperatures. In this work, I propose a mechanism to account for this experimental result. I use the localization of chiral Majorana modes in defects to explain the presence of gapless modes in the bulk. Treating the weak interaction between the chiral modes in the defects within a mean-field approximation, I find that these modes remain gapless when the interaction is below a critical value that depends on the magnetic field strength. I used the effective low-energy theory to calculate the spin-lattice relaxation rate and found a behavior that agrees with the experimental result.
Topologically ordered states are quantum states of matter with topological ground-state degeneracy and quasiparticles carrying fractional quantum numbers and fractional statistics. The topological spin ${\ensuremath{\theta}}_{a}=2\ensuremath{\pi}{h}_{a}$ is an important property of a topological quasiparticle, which is the Berry phase obtained in the adiabatic self-rotation of the quasiparticle by $2\ensuremath{\pi}$. For chiral topological states with robust chiral edge states, another fundamental topological property is the edge state chiral central charge ccc. In this paper we propose an approach to compute the topological spin and chiral central charge in lattice models by defining a quantity that we call the momentum polarization. Momentum polarization is defined on the cylinder geometry as a universal subleading term in the average value of a ``partial translation operator.'' We show that the momentum polarization is a quantum entanglement property which can be computed from the reduced density matrix, and our analytic derivation based on edge conformal field theory shows that the momentum polarization measures the combination ${h}_{a}\ensuremath{-}\frac{c}{24}$ of topological spin and central charge. Results are obtained for two example systems, the non-Abelian phase of the honeycomb lattice Kitaev model and the $\ensuremath{\nu}=1/2$ Laughlin state of a fractional Chern insulator described by a variational Monte Carlo wave function, which verify the analytic formula with high accuracy and further suggest that this result remains robust even when the edge states cannot be described by a conformal field theory. Our result provides an efficient approach to extract characteristic quantities of topological states of matter from finite size numerics.
Quantum spin liquids, a highly topologically entangled, dynamically correlated state where quantum fluctuations preclude any long-range ordering down to absolute zero. In the search for a topologically robust qubit, the scientific community has been in continuous hunt for real quantum spin liquid systems. Alexei Kitaev in his exactly solvable model for a spin-1/2 two-dimensional honeycomb lattice, presented a system that hosts a topologically protected state (Majorana zero-modes). Under an applied external field, the Kitaev spin liquids turn into a topologically non-trivial chiral spin-liquid state with non-abelian anionic excitations, which is crucial for quantum computing. Earlier theoretical predictions advocated that Kitaev physics can be realized in spin-orbit-coupled Mott insulators such as honeycomb irradiates and ruthenates. However, the experimental findings continuously challenge the theoretical aspects, indicating the presence of non-Kitaev interactions in real materials, where the dimensionality, disorder (vacancy), chemical composition, generalized spin-S, and external perturbations (pressure, magnetic field, temperature) can actively tune the Kitaev interactions and the ground state excitations. In this review article, a comprehensive discussion is included with an updated literature survey in the context of the potential of Raman spectroscopy as a probe for Kitaev quantum spin liquids.
The chiral central charge c−c_-c−​ is a key topological invariant of the edge characterizing the bulk two-dimensional chiral topological order, but its direct evaluation in microscopic spin models has long been a challenge, especially for non-abelian topological order. Building on the recently developed modular commutator formalism, we numerically obtain c−c_-c−​ directly from single ground-state wave functions of two-dimensional interacting spin models that have chiral topological order. This provides a geometry-independent and bulk diagnostic of chirality. We study two nonintegrable systems -- the Zeeman-Kitaev honeycomb model and the kagome antiferromagnet -- both subjected to scalar spin chirality perturbations. We find that the modular commutator yields results consistent with the expected topological quantum field theories. We also compute the topological entanglement entropy which provides an independent diagnostic of the topological orders. Our work establishes modular commutators as a powerful numerical probe of chiral topological order in strongly correlated quantum magnets.