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arXiv:2608.06098

An Effective String Theory Toolbox for Quantum Hall Interfaces II: Majorana Fermions on Fluctuating Moore-Read Worldsheets

Ken K. W. Ma

cond-mat.mes-hallcond-mat.str-elhep-thquant-phSubmitted Aug 6, 2026• Announced Aug 7, 2026
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Abstract

A Moore--Read interface carries a chiral Majorana mode on a boundary whose geometry may itself fluctuate. Fixed-edge theory does not determine how this neutral mode should be transported when the interface bends and moves, or how its dynamics couples to the fluctuating shape. Here we construct a spatially reparametrization-invariant Majorana theory on the nonrelativistic worldsheet of a freely moving interface. The changing line element fixes a universal half-density transport law, while additional curvature- and velocity-dependent couplings remain controlled by microscopic interface physics. The resulting framework identifies the Majorana stress tensor as the mediator between neutral and geometric dynamics and provides the neutral sector needed for effective theories of dynamical non-Abelian quantum Hall interfaces.

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•May 1, 2026•0 citations
A Moore--Read interface carries a chiral Majorana mode on a boundary whose geometry may itself fluctuate. Fixed-edge theory does not determine how this neutral mode should be transported when the interface bends and moves, or how its dynamics couples to the fluctuating shape. Here we construct a spatially reparametrization-invariant Majorana theory on the nonrelativistic worldsheet of a freely moving interface. The changing line element fixes a universal half-density transport law, while additional curvature- and velocity-dependent couplings remain controlled by microscopic interface physics. The resulting framework identifies the Majorana stress tensor as the mediator between neutral and geometric dynamics and provides the neutral sector needed for effective theories of dynamical non-Abelian quantum Hall interfaces.
A freely moving quantum Hall interface is fundamentally different from an ordinary edge fixed by an external confining potential. Since a normal displacement changes the areas occupied by the adjacent incompressible phases, the interface geometry and charge dynamics cannot be treated as independent degrees of freedom. We formulate this problem for interfaces between Abelian quantum Hall phases using a spatially reparametrization-invariant worldsheet description, in which tangential motion is a relabeling of the interface while normal motion is physical. Starting from the two-sided Chern--Simons response, we derive the relation between normal charge transport and interface motion. We then introduce a relative-area construction, defined with respect to a material reference curve, that converts this velocity relation into an equal-time constraint linking the charged boundary sector to the interface shape. Combined with the folded KKK-matrix current algebra, this identifies the universal Hall kinematics of the moving interface while leaving its geometric energy and neutral dynamics dependent on microscopic interface physics. The resulting framework provides a systematic basis for effective theories of dynamical quantum Hall interfaces.
A freely moving quantum Hall (QH) interface may end on a physical edge or topological boundary, but fixed-edge theory cannot determine what endpoint data make such a termination consistent. Here we formulate an open-worldsheet junction framework in which the embedding, material charge, anomaly flow, and topological boundary condition are organized together. The endpoint is specified by a geometric support and variational boundary data, together with condensable topological sectors and any outgoing channels required to absorb or continue the worldsheet flux. This construction extends the charge--shape relation to an interval and shows why a lone chiral Majorana cannot terminate on a finite-dimensional endpoint degree of freedom. It gives an operational definition of a QH brane and a systematic basis for endpoint and network theories of dynamical QH interfaces.
We consider an interface separating the Moore-Read state and Halperin 331 state in a half-filled Landau level, which can be realized in a double quantum well system with varying interwell tunneling and/or interaction strengths. In the presence of electron tunneling and strong Coulomb interactions across the interface, we find that all charge modes localize and the only propagating mode left is a chiral Majorana fermion mode. Methods to probe this neutral mode are proposed. A quantum phase transition between the Moore-Read and Halperin 331 states is described by a network of such Majorana fermion modes. In addition to a direct transition, they may also be separated by a phase in which the Majorana fermions are delocalized, realizing an incompressible state which exhibits quantum Hall charge transport and bulk heat conduction.
Quantum Hall bilayers are a uniquely tunable platform that can realize continuous transitions between distinct topological phases of matter. One prominent example is the transition between the Halperin state and the Moore-Read Pfaffian, long predicted to host a critical theory of Majorana fermions but so far not verified in unbiased microscopic simulations. Using the fuzzy-sphere regularization, we identify the low-energy spectrum at this transition with the 3D gauged Majorana conformal field theory. We show that the transition is driven by the closing of the neutral fermion gap, and we directly extract the operator content in both integer and half-integer spin sectors. Our results resolve the long-standing question of the nature of a topological phase transition in a setting relevant to quantum Hall experiments, while also providing a realization of emergent fermionic fields on the fuzzy sphere, previously limited to bosonic fields.
Co-propagating chiral Majorana edge modes are predicted to exist in proximity-coupled quantum anomalous Hall systems. They combine the exciting prospects of edge vortices as flying non-Abelian Ising anyons, with their fully electrical detection through anyon fusion processes. In this setup, electrical transport arises from the interference of a pair of Majorana edge states. By bosonizing the model with an additional replica sector, we generalize previous treatments to fully asymmetric configurations. We analyze a Mach-Zehnder interferometer and show that while the DC conductance does not exhibit signatures of edge-vortex interference, at low frequencies, this interference manifests as an effective capacitance. This quantity is sensitive to the non-Abelian anyon statistics of edge vortices and, in particular, carries signatures of their topological spin and nontrivial conformal dimension. We demonstrate that the thermal length lT=ℏv/(πkBT)l_T = \hbar v/(πk_B T)lT​=ℏv/(πkB​T) is the relevant scale at low temperatures TTT, analogous to quasiparticle transport in conventional quantum Hall edge states with velocity vvv. As long as the interferometer arm lengths match within this length scale, transport signatures of the non-Abelian statistics of edge vortices remain visible.
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.