R. Jackiw
Dedicated to Ludwig Faddeev on his 80th birthday. Ludwig exemplifies perfectly a mathematical physicist: significant contribution to mathematics (algebraic properties of integrable systems) and physics (quantum field theory). In this note I present an exercise which bridges mathematics (restricted Clifford algebra) to physics (Majorana fermions).
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To mark the 111th birthday of Eugene Wigner, we review topological excitations in diverse dimensions.
The Structure of the Majorana Clifford Group
本文研究马约拉纳费米子系统中的克利福德算子类似物，证明宇称守恒的马约拉纳克利福德子群可由二元域正交群表示，能通过编织算子生成并构造偶宇称稳定子码，且其框架势与普通克利福德群等价。
We show that a strongly interacting chain of Majorana zero modes exhibits a supersymmetric quantum critical point corresponding to the $c={7\over 10}$ tricritical Ising model, which separates a critical phase in the Ising universality class from a supersymmetric massive phase. We verify our predictions with numerical density-matrix-renormalization-group computations and determine the consequences for tunnelling experiments.
In this paper we study a Clifford algebra generalization of the quaternions and its relationship with braid group representations related to Majorana fermions. The Fibonacci model for topological quantum computing is based on the fusion rules for a Majorana fermion. Majorana fermions can be seen not only in the structure of collectivies of electrons, as in the quantum Hall effect, but also in the structure of single electrons both by experiments with electrons in nanowires and also by the decomposition of the operator algebra for a fermion into a Clifford algebra generated by two Majorana oper…
Emergent quantum Majorana metal from a chiral spin liquid
本文提出在外加磁场下反铁磁Kitaev模型中π-通量增殖导致马约拉纳零模重叠，从而形成具有零能费米面的Z₂量子马约拉纳金属态，并用马约拉纳谱函数验证了该变分图像。
Majorana fermions cannot be found in nature as a free fundamental particle. Nevertheless, in condensed matter systems, they can emerge as a collective excitation. In this work, using functional integration techniques, we calculated the effective potential for emergent Majorana fermions in the Kitaev chain. In this case, we have shown how the superconductor parameter behaves as a function of temperature. Furthermore, we considered surface-induced superconductivity in a Topological Insulator and calculated the effective potential for emergent Majorana fermions in this system. In the case of an s…
We exhibit an exactly solvable example of a SU(2) symmetric Majorana spin liquid phase, in which quenched disorder leads to random-singlet phenomenology. More precisely, we argue that a strong-disorder fixed point controls the low temperature susceptibility $χ(T)$ of an exactly solvable $S=1/2$ model on the decorated honeycomb lattice with quenched bond disorder and/or vacancies, leading to $χ(T) = {\mathcal C}/T+ {\mathcal D} T^{α(T) - 1}$ where $α(T) \rightarrow 0$ as $T \rightarrow 0$. The first term is a Curie tail that represents the emergent response of vacancy-induced spin textures spre…
Dihedral twist liquid models from emergent Majorana fermions
本文提出了一族基于电子的耦合线模型构建二维玻色扭曲液体拓扑相，通过将演生马约拉纳费米子的二面体对称性提升为规范对称性，构建了精确可解模型并分析了任意子激发的统计性质。
Topological states of matter present a wide variety of striking new phenomena. Prominent among these is the fractionalisation of electrons into unusual particles: Majorana fermions [1], Laughlin quasiparticles [2] or magnetic monopoles [3]. Their detection, however, is fundamentally complicated by the lack of any local order, such as, for example, the magnetisation in a ferromagnet. While there are now several instances of candidate topological spin liquids [4], their identification remains challenging [5]. Here, we provide a complete and exact theoretical study of the dynamical structure fact…
The vortex of iron-based superconductors is emerging as a promising platform for Majorana zero mode, owing to a magic integration among intrinsic vortex winding, non-trivial band topology, strong electron-electron correlations, high-Tc superconductivity and the simplification of single material. It overcomes many difficulties suffered in heterostructure-based Majorana platforms, including small topological gap, interfacial contamination, lattice imperfections, and etc. Isolated zero-bias peaks have been found in vortex of several iron-based superconductors. So far, studies from both experiment…
Emergent Andreev Reflection from a Lattice Duality Defect
本文证明，横向场伊辛链的晶格对偶缺陷可产生类似安德烈夫反射的电荷共轭边界条件，为双通道近藤问题和Maldacena-Ludwig单极子散射提供了微观解释。
Qubit fractionalization and emergent Majorana liquid in the honeycomb Floquet code induced by coherent errors and weak measurements
通过调节蜂巢Floquet码的测量强度并引入相干错误，揭示了量子比特分数化、马约拉纳液体形成及配对有序三个阶段，证明其弱测量下保持长程纠缠而非崩溃。
Is chiral supersymmetry emanant or emergent?
该研究探讨扩展Majorana Nicolai模型中手征超对称的涌现与显现性质，发现自发破缺导致低能自由度加倍，并归因于涌现手征超对称的破缺。
Whereas identification of Cooper pair symmetry is the first and crucial step in the investigation of unconventional superconductors, only a few have been established so far because of its own difficulties. To solve this problem, we develop a theory for identification of pairing symmetry using knowledge of topological superconductivity. Establishing the multipole theory of emergent Majorana fermions in time-reversal-invariant topological superconductors, we discover a one-to-one correspondence between the electromagnetic response of Majorana fermions and Cooper pair symmetry. The emergent Major…
Floquet Majorana flat bands and emergent Cooper pair symmetries in $p-$wave magnet$-$superconductor heterostructure
研究$p$-波磁体与$s$-波超导体界面拓扑超导性，发现七种节点拓扑相，具有Majorana零能平带和量子化零偏压电导峰，并揭示周期驱动产生Floquet Majorana平带及新型非平衡配对通道。
We theoretically address spin chain analogs of the Kitaev quantum spin model on the honeycomb lattice. The emergent quantum spin liquid phases or Anderson resonating valence bond (RVB) states can be understood, as an effective model, in terms of p-wave superconductivity and Majorana fermions. We derive a generalized phase diagram for the two-leg ladder system with tunable interaction strengths between chains allowing us to vary the shape of the lattice (from square to honeycomb ribbon or brickwall ladder). We evaluate the winding number associated with possible emergent (topological) gapless m…
Quantum matter provides an effective vacuum out of which arise emergent particles not corresponding to any experimentally detected elementary particle. Topological quantum materials in particular have become a focus of intense research in part because of the remarkable possibility to realize Majorana fermions, with their potential for new, decoherence-free quantum computing architectures. In this paper we undertake a study on high-quality single crystal of $α-RuCl_3$ which has been identified as a material realizing a proximate Kitaev state, a topological quantum state with magnetic Majorana f…
Landau levels (LL) have been predicted to emerge in systems with Dirac nodal points under applied non-uniform strain. We consider 2D, $d_{xy}$ singlet (2D-S) and 3D $p \pm i p$ equal-spin triplet (3D-T) superconductors (SCs). We demonstrate the spinful Majorana nature of the bulk gapless zeroth-LLs. Strain along certain directions can induce two topologically distinct phases in the bulk, with zeroth LLs localized at the the interface. These modes are unstable toward ferromagnetism for 2D-S cases. Emergent real-space Majorana fermions in 3D-T allow for more exotic possibilities.
Spin liquids may host emergent quasiparticles, collective excitations of the spin degrees of freedom with characteristic features of Majorana fermions, which experimentally are detectable by broad excitation continua due to spin fractionalization. The latter is predicted for the Kitaev spin liquid, an exactly solvable model with bond-dependent interactions on a two-dimensional honeycomb lattice. Here we report on detailed THz experiments in α-RuCl3, identifying these characteristic fingerprints of Majorana fermions. The continuum intensity decreases and finally vanishes on increasing temperatu…
Electromagnetic response of spinful Majorana fermions
该研究针对拓扑超导体中有自旋马约拉纳费米子的电磁响应展开，构建了通用理论阐明配对对称性的编码方式，并预测了铁基超导体角态马约拉纳费米子的子晶格依赖偶极型磁响应。
The Weyl fermions with a well defined chirality are known to demand that the dimension of space which they inhabit must be odd. It is shown here, however, that not all odd dimensional spaces are equally good hosts: in particular, an arbitrary number of chiral Weyl fermions can acquire a Majorana mass only in three (modulo eight) dimensions. The argument utilizes a) the precise analogy that exists between the Majorana mass term and the Cooper pairing of time-reversed Weyl fermions, and b) the conditions on the requisite time-reversal operator, which are implied by the Clifford algebra. The theo…
We construct a dual bosonized description of a massless Majorana fermion in $(2+1)d$. In contrast to Dirac fermions, for which a bosonized description can be constructed using a flux attachment procedure, neutral Majorana fermions call for a different approach. We argue that the dual theory is an $SO(N)_1$ Chern-Simons gauge theory with a critical $SO(N)$ vector bosonic matter field ($N \geq 3$). The monopole of the $SO(N)$ gauge field is identified with the Majorana fermion. We provide evidence for the duality by establishing the correspondence of adjacent gapped phases and by a parton constr…
We investigate the admittance of a metallic quantum RC circuit with a spinful single-channel lead or equally with two conducting spin-polarized channels, in which Majorana fermions play a crucial role in the charge dynamics. We address how the two-channel Kondo physics and its emergent Majoranas arise. The existence of a single unscreened Majorana mode results in non-Fermi-liquid features and we determine the universal crossover function describing the Fermi-liquid to non-Fermi-liquid region. Remarkably, the same universal form emerges both at weak transmission and large transmission. We find …
Mechanical strain can generate a pseudo-magnetic field, and hence Landau levels (LL), for low energy excitations of quantum matter in two dimensions. We study the collective state of the fractionalised Majorana fermions arising from residual generic spin interactions in the central LL, where the projected Hamiltonian reflects the spin symmetries in intricate ways: emergent U(1) and particle-hole symmetries forbid any bilinear couplings, leading to an intrinsically strongly interacting system; also, they allow the definition of a filling fraction, which is fixed at 1/2. We argue that the result…
We study the effects of the flux configurations on the emergent Majorana fermions in the $S=1/2$ Kitaev model on a honeycomb lattice, where quantum spins are fractionalized into itinerant Majorana fermions and localized fluxes. A quantum spin liquid appears as the ground state of the Kitaev model in the flux-free sector, which has intensively been investigated so far. In this flux sector, the Majorana fermion system has linear dispersions and shows power law behavior in the Majorana correlations. On the other hand, periodically-arranged flux configurations yield low-energy excitations in the M…
In this article, we discuss strong coupling limits of topological quantum critical points (TQCPs) where quantum phase transitions between two topological distinct superconducting states take place. We illustrate that while superconducting phases on both sides of TQCPs spontaneously break same symmetries, universality classes of critical states can be identified only when global symmetries in topological states are further specified. In dimensions $d=2,3$, we find that continuous $(d+1)$th order transitions at weakly interacting TQCPs that were pointed out previously in the presence of emergent…
Majorana fermions are the real (in a mathematical sense) counterparts of complex fermions like ordinary electrons. The promise of topological quantum computing has lead to substantial experimental progress in realizing these particles in various synthetic platforms. The realization of Majorana fermions motivates a fundamental question: What phases of matter can emerge if many Majorana fermions are allowed to interact? Here we review recent progress in this direction on the proposed experimental setups, analytical and numerical results on low-dimensional lattice models, and the exactly solvable…
Majorana fermions were originally proposed as elementary particles acting as their own antiparticles. In recent years, it has become clear that Majorana fermions can instead be realized in condensed-matter systems as emergent quasiparticles, a situation often accompanied by topological order. Here we propose a physical system which realizes Landau levels - highly degenerate single-particle states usually resulting from an orbital magnetic field acting on charged particles - for Majorana fermions. This is achieved in a variant of a quantum spin system due to Kitaev which is distorted by triaxia…
C-R-T Fractionalization in the First Quantized Hamiltonian Theory
本文研究了费米子CRT对称性的分数化现象，揭示了费米子CRT对称性与标量玻色子CRT对称性的区别，并通过引入内对称性如费米子奇偶性，扩展了CRT对称性。
Magnetic field induced phenomena in Kitaev spin liquids
研究磁场对Kitaev自旋液体中分数化准粒子响应的影响，揭示实验识别方法及新兴相的条件。
Free and Interacting Fermionic Conformal Field Theories on the Fuzzy Sphere
本文研究在模糊球上自由和相互作用费米共形场论，通过局部费米算子和角动量不匹配实现相变，并发现超Ising理论的超共形对称性。
We study the quantum multicritical point in a (2+1)-dimensional Dirac system between the semimetallic phase and two ordered phases that are characterized by anticommuting mass terms with $O(N_1)$ and $O(N_2)$ symmetry, respectively. Using $ε$ expansion around the upper critical space-time dimension of four, we demonstrate the existence of a stable renormalization-group fixed point, enabling a direct and continuous transition between the two ordered phases directly at the multicritical point. This point is found to be characterized by an emergent $O(N_1+N_2)$ symmetry for arbitrary values of $N…
We study an interface between a Kitaev spin liquid (KSL) in the chiral phase and a non-chiral superconductor. When the coupling across the interface is sufficiently strong, the interface undergoes a transition into a phase characterized by a condensation of a bound state of a Bogoliubov quasiparticle in the superconductor and an emergent fermionic excitation in the spin liquid. In the condensed phase, electrons in the superconductor can coherently convert into emergent fermions in the spin liquid and vice versa. As a result, the chiral Majorana edge mode of the spin liquid becomes visible in t…
By using the variational matrix product state method, we numerically study the interacting Kitaev chain with spatially varying periodic and quasi-periodic potentials and the latter follows the Fibonacci sequence. The edge correlation functions of Majorana fermions and low-lying ground states are computed to explore the robustness of topological superconducting phase. It is found that the original topological nontrivial phase is separated into to two branches by an emergent topological trivial phase, as a result of the competition among spatially varying potential, electronic Coulomb interactio…
Motivated by recent synthesis of the hyper-honeycomb material $β$-$\mathrm{Li_2 Ir O_3}$, we study the dynamical structure factor (DSF) of the corresponding 3D Kitaev quantum spin-liquid (QSL), whose fractionalised degrees of freedom are Majorana fermions and emergent flux-loops. Properties of this 3D model are known to differ in important ways from those of its 2D counterpart -- it has finite-temperature phase transition, as well as distinct features in Raman response. We show, however, that the qualitative behaviour of the DSF is broadly dimension-independent. Characteristics of the 3D DSF i…
We study multiband semiconducting nanowires proximity-coupled with an s-wave superconductor and calculate the topological phase diagram as a function of the chemical potential and magnetic field. The non-trivial topological state corresponds to a superconducting phase supporting an odd number of pairs of Majorana modes localized at the ends of the wire, whereas the non-topological state corresponds to a superconducting phase with no Majoranas or with an even number of pairs of Majorana modes. Our key finding is that multiband occupancy not only lifts the stringent constraint of one-dimensional…
Soliton-antisoliton pairs in the supersymmetric gapped phase of an interacting Majorana chain
研究强相互作用马约拉纳链中超对称三临界伊辛相在有能隙相中的表现，发现孤子-反孤子对作为最低激发态，每个孤子束缚一个局域马约拉纳模，形成非局域狄拉克费米子。
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