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Majorana physics in entangled quantum matter: from Kitaev spin liquids to monitored quantum circuits

Kai Klocke

Published May 1, 2026• 0 citations
OpenAlex

Abstract

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.

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Quantum circuits offer a versatile platform for simulating digital quantum dynamics and uncovering novel states of nonequilibrium quantum matter. One principal example are measurement-induced phase transitions arising from nonunitary dynamics in monitored circuits, which employ midcircuit measurements as an essential building block next to standard unitary gates. Although a comprehensive understanding of the dynamics in generic circuits is still evolving, we contend that monitored quantum circuits give rise to robust phases of dynamic matter, which---akin to Hamiltonian ground-state phases---yield emergent universal behavior, which can be categorized based on circuit symmetries and spatial dimensionality. To illustrate this concept, we focus on measurement-only quantum circuits within symmetry classes BDI and D, which are measurement-only circuit adaptations of the paradigmatic Kitaev and Yao-Kivelson models, embodying particle-hole-symmetric Majorana fermions with or without time reversal. We establish a general framework (Majorana loop models) for both symmetry classes (in arbitrary spatial dimensions) to provide access to the phenomenology of the entanglement dynamics in these circuits, displaying both an area-law phase of localized Majorana loops and a delocalized, highly entangled Majorana liquid phase. The two phases are separated by a continuous transition displaying quantum Lifshitz scaling, albeit with critical exponents of two distinct, non-Hamiltonian universality classes. The loop model framework provides not only analytical understanding of these universality classes in terms of nonlinear sigma models but also allows for highly efficient numerical techniques capable of simulating excessively large circuits with up to ${10}^{8}$ qubits. We utilize this framework to accurately determine universal probes that distinguish both the entangled phases and the critical points of the two symmetry classes. Our work thereby further solidifies the concept of emergent circuit phases and their phase transitions.

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arXiv:2602.09709•Feb 10, 2026•0 citations
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.
Quantum circuits offer a versatile platform for simulating digital quantum dynamics and uncovering novel states of nonequilibrium quantum matter. One principal example are measurement-induced phase transitions arising from nonunitary dynamics in monitored circuits, which employ midcircuit measurements as an essential building block next to standard unitary gates. Although a comprehensive understanding of the dynamics in generic circuits is still evolving, we contend that monitored quantum circuits give rise to robust phases of dynamic matter, which---akin to Hamiltonian ground-state phases---yield emergent universal behavior, which can be categorized based on circuit symmetries and spatial dimensionality. To illustrate this concept, we focus on measurement-only quantum circuits within symmetry classes BDI and D, which are measurement-only circuit adaptations of the paradigmatic Kitaev and Yao-Kivelson models, embodying particle-hole-symmetric Majorana fermions with or without time reversal. We establish a general framework (Majorana loop models) for both symmetry classes (in arbitrary spatial dimensions) to provide access to the phenomenology of the entanglement dynamics in these circuits, displaying both an area-law phase of localized Majorana loops and a delocalized, highly entangled Majorana liquid phase. The two phases are separated by a continuous transition displaying quantum Lifshitz scaling, albeit with critical exponents of two distinct, non-Hamiltonian universality classes. The loop model framework provides not only analytical understanding of these universality classes in terms of nonlinear sigma models but also allows for highly efficient numerical techniques capable of simulating excessively large circuits with up to 108{10}^{8}108 qubits. We utilize this framework to accurately determine universal probes that distinguish both the entangled phases and the critical points of the two symmetry classes. Our work thereby further solidifies the concept of emergent circuit phases and their phase transitions.
Quantum spin liquids realize massive entanglement and fractional quasiparticles from localized spins, proposed as an avenue for quantum science and technology. In particular, topological quantum computations are suggested in the non-abelian phase of Kitaev quantum spin liquid with Majorana fermions, and detection of Majorana fermions is one of the most outstanding problems in modern condensed matter physics. Here, we propose a concrete way to identify the non-abelian Kitaev quantum spin liquid by magnetic field angle dependence. Topologically protected critical lines exist on a plane of magnetic field angles, and their shapes are determined by microscopic spin interactions. A chirality operator plays a key role in demonstrating microscopic dependences of the critical lines. We also show that the chirality operator can be used to evaluate topological properties of the non-abelian Kitaev quantum spin liquid without relying on Majorana fermion descriptions. Experimental criteria for the non-abelian spin liquid state are provided for future experiments.
Topological quantum computing seeks to store and manipulate information in a protected manner using topological phases of matter. Information encoded in the degenerate state space of pairs of non-Abelian anyons or defects is robust to local perturbations, reducing its susceptiblity to environmental errors and potentially providing a scalable approach to quantum computing. However, topological quantum computing faces significant challenges, not least of which is identifying an experimentally accessible platform supporting non-Abelian topological physics. In this thesis, we critically analyze topological quantum computing with Majorana zero modes, non-Abelian defects of a topological superconductor. We identify intrinsic error sources for Majorana-based systems and propose quantum computing architectures that minimize their effects. Additionally, we consider a new approach for realizing and detecting non-Abelian topological defects in fractional Chern insulators. Topological quantum computing is predicated on the idea that braiding non-Abelian anyons adiabatically can implement quantum gates fault tolerantly. However, any braiding experiment will necessarily depart from the strict adiabatic limit. We begin by analyzing the nature of diabatic errors for anyon braiding, paying particular attention to how such errors scale with braiding time. We find that diabatic errors are unfavorably large and worryingly sensitive to details of the time evolution. We present a measurement-based correction protocol for such errors, and illustrate its application in a particular Majorana-based qubit design. We next propose designs for Majorana-based qubits operated entirely by a measurement-based protocol, thereby avoiding the diabatic errors discussed above. Our designs can be scaled into large two dimensional arrays amenable to long-term quantum computing goals, whose core components are testable in near-term devices. These qubits are robust to quasiparticle poisoning, anticipated to be one of the dominant error sources coupling to Majorana zero modes. We demonstrate that our designs support topologically protected Clifford operations and can be augmented to a universal gate set without requiring additional control parameters.While topological protection greatly suppresses errors, residual coupling to noise limits the lifetimes of our proposed Majorana-based qubits. We analyze the dephasing times for our quasiparticle-poisoning-protected qubits by calculating their charge distribution using a particle number-conserving formalism. We find that fluctuations in the electromagnetic environment couple to an exponentially suppressed topological dipole moment. We estimate dephasing times due to 1/f1/f1/f noise, thermal quasiparticle excitations, and phonons for different qubit sizes.The residual errors discussed above will necessarily require error correction for a sufficiently long quantum computation. We develop physically motivated noise models for Majorana-based qubits that can be used to analyze the performance of a quantum error correcting code. We apply this noise model to estimate pseudo-thresholds for a small subsystem code, identifying the relative importance of difference error processes from a fault tolerance perspective. Our results emphasize the necessity of suppressing long-lived quasiparticle excitations that can spread across the code.Finally, we turn our attention to a different platform that could host non-Abelian topological defects: fractional Chern insulators in graphene. We study the edge states of fractional Chern insulators using the field theory of fractional quantum Hall edges supplemented with a symmetry action. We find that lattice symmetries impose a quantized momentum difference for edge electrons in a fractional state of a C=2C=2C=2 Chern band. This momentum difference can be used to selectively contact the different edge states, thereby allowing detection of topological defects in the bulk with a standard four terminal measurement. Our proposal could be implemented in graphene subject to an artificially patterned lattice.
Recent decades have seen a proliferation of unconventional quasiparticles in condensed matter systems. Majorana fermions, theoretically predicted in several setups and of great interest in topological quantum computation, are a focus of intense research efforts. Higher-spin moments, relevant for both solid-state compounds and cold atoms, are of great theoretical interest as they interpolate between the quantum and classical limits but sometimes show surprising behavior. In the meantime, successful fabrication and characterization of low-dimensional systems have brought new phenomenology and physical insights. In this dissertation I will theoretically explore a few low-dimensional models using Majorana fermions and higher-spin moments as building blocks. I will first discuss a generalized family of the celebrated Sachdev-Ye-Kitaev model, a zero-dimensional all-to-all Majorana model that exhibits non-Fermi liquid behavior and is holographically dual to a black hole. The generalized model has a phase transition between a non-Fermi liquid and a disordered Fermi liquid. Then I will discuss the Heisenberg model with higher spins, with a focus on chaos and information scrambling. Using matrix-product-state-based methods, we are able to obtain numerical results for spin up to 4 and characterize the Lyapunov growth. After that I will discuss a generalization of the Hubbard model to Majorana fermions on the honeycomb lattice. Unlike previous similar models, we find topological phases with (anti-)chiral edge modes for weak interaction. Finally I will show a construction of explicit supersymmetric Majorana model on the kagome lattice, where a family of exact solutions is found and the nature of supersymmetry breaking is explored.
The fractionalization of quantum numbers in interacting quantum many-body systems is a central motif in condensed-matter physics with prominent examples including the fractionalization of the electron in quantum Hall liquids or the emergence of magnetic monopoles in spin-ice materials. Here, we discuss the fractionalization of magnetic moments in three-dimensional Kitaev models into Majorana fermions (and a Z_{2} gauge field) and their emergent collective behavior. We analytically demonstrate that the Majorana fermions form a Weyl superconductor for the Kitaev model on the recently synthesized hyperhoneycomb structure of β-Li_{2}IrO_{3} when applying a magnetic field. We characterize the topologically protected bulk and surface features of this state, which we dub a Weyl spin liquid, including thermodynamic and transport signatures.
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.