Chang-Tse Hsieh, Yu Nakayama, Yuji Tachikawa
We show that there is a fermionic minimal model, i.e. a 1+1d conformal field theory which contains operators of half-integral spins in its spectrum, for each $c=1-6/m(m+1)$, $m\ge 3$. This generalizes the Majorana fermion for $c=1/2$, $m=3$ and the smallest $\mathcal{N}{=}1$ supersymmetric minimal model for $c=7/10$, $m=4$. We provide explicit Hamiltonians on Majorana chains realizing these fermionic minimal models.
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We study free fermionic models that host Majorana zero modes using the Majorana orthogonal transformation, which is a type of transformation between different fermionic models under Majorana representation of complex fermions. Using Majorana orthogonal transformation, a U(1) topological gauge theory for the doubled $p_{x}+ip_{y}$ topological superconductor is obtained; the vortex Majorana zero modes and the degeneracy splitting of multiple vortices are studied using field theoretical method. For lattice Majorana hopping models, we perform real-space analysis on the Majorana zero modes. In one …
Majorana-fermionic quantum computation (MFQC) was proposed by Bravyi and Kitaev (See Ref.\cite{Kitaev}), in which a fault-torrent (non-topological) quantum computer built from Majorana fermions may be more efficient than that built from distinguishable two-state systems. However, till now people don't know how to realize a MFQC in a physical system. In this paper we proposed a possible realization of MFQC. We find that the end of a line-defect of p-wave superconductor or superfluid on a honeycomb lattice will trap a Majorana zero mode, which will become the starting point of MFQC. Then we show…
Fermionic influence superoperator for transport through Majorana zero modes
本研究推导了描述电子穿越马约拉纳零模开放量子动力学的费米子影响超算符，构建层级运动方程实现数值计算，提出泛函微分方案给出输运可观测量精确表达式，为马约拉纳物理输运特征分析奠定理论基础。
Majorana-Pauli stabilizer codes and duality webs of fermionic topological phases
本文引入马约拉纳-泡利稳定子码，构造了费米子环面码的精确可解稳定子实现，并通过任意子凝聚和对称性规范对偶网络连接了多种拓扑相。
Spontaneous Supersymmetry Breaking and Nambu-Goldstone Modes in Interacting Majorana Chains
本文研究相互作用马约拉纳链中的自发超对称破缺，引入具有SUSY的模型并分析其产生的南部-戈德斯通模式，以弥补非相对论系统中该问题研究不足的空白。
We devise a number-conserving scheme for the realization of Majorana Zero Modes in an interacting fermionic ladder coupled by Aharonov-Bohm cages. The latter provide an efficient mechanism to cancel single-particle hopping by destructive interference. The crucial parity symmetry in each wire is thus encoded in the geometry of the setup, in particular, its translation invariance. A generic nearest-neighbor interaction generates the desired correlated hopping of pairs. We exhibit the presence of an extended topological region in parameter space, first in a simplified effective model via bosoniza…
We initiate the study of Majorana fermion codes. These codes can be viewed as extensions of Kitaev's 1D model of unpaired Majorana fermions in quantum wires to higher spatial dimensions and interacting fermions. The purpose of Majorana fermion codes (MFCs) is to protect quantum information against low-weight fermionic errors, that is, operators acting on sufficiently small subsets of fermionic modes. We examine to what extent MFCs can surpass qubit stabilizer codes in terms of their stability properties. A general construction of 2D MFCs is proposed which combines topological protection based …
In this short note, we comment on the existence of two more fermionic unitary minimal models not included in recent work by Hsieh, Nakayama, and Tachikawa. These theories are obtained by fermionizing the $\mathbb{Z}_2$ symmetry of the m=11 and m=12 exceptional unitary minimal models. Furthermore, these should be the only missing cases.
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 a chain of ferromagnetic nano-particles or ferromagnetic molecule/atoms on a substrate of fully gapped superconductors. We find that under quite realistic conditions, the fermion-number-parity symmetry $Z_2^f$ can spontaneously break. In other words, such a chain can realize a 1+1D fermionic topologically ordered state and the corresponding two-fold topological degeneracy on an open chain. Such a topological degeneracy becomes the so called Majorana zero mode in the non-interacting limit. More specifically, we find that $Z_2^f$ symmetry breaking or fermionic 1+1D topological order can…
We describe ground states of correlated electron systems in which the electron fractionalizes into separate quasiparticles which carry its spin and its charge, and into real Majorana fermions which carry its Fermi statistics. Such parent states provide a unified theory of previously studied fractionalized states: their descendants include insulating and conducting states with neutral spin S=1/2 fermionic spinons, and states with spinless fermionic charge carriers. We illustrate these ideas on the honeycomb lattice, with field theories of such states and their phase transitions.
Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
本研究采用拉回平凡化技术扩展费米子对称性保护拓扑态的对称群，系统构建其有能隙界面与反常边界，推导了低维系统的一致性公式，并给出了具体的三维示例。
Under an appropriate symmetric bulk bipartition in a one-dimensional symmetry protected topological phase with the Affleck-Kennedy-Lieb-Tasaki matrix product state wave function for the odd integer spin chains, a bulk critical entanglement spectrum can be obtained, describing the excitation spectrum of the critical point separating the topological phase from the trivial phase with the same symmetry. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry breaking phase transitions. Recently, the framework of matrix product states for topological phases with Ma…
Quantized Stabilizer-Rényi Boundary Response across Fermionic SPT Transitions
该研究针对费米子SPT相变，在α=1/2时分离出体扣除后的边界响应，其值与绕数变化相关，可计数改变费米子SPT指数的马约拉纳通道。
Motivated by recent experiments in the Kitaev honeycomb lattice, Kondo insulators, and the "Luttinger's theorem-violating" Fermi liquid phase of the underdoped cuprates, we extend the theoretical machinery of Landau-Fermi liquid theory to a system of itinerant, interacting Majorana-like particles. Building upon a previously introduced model of "nearly self-conjugate" fermionic polarons, a Landau-Majorana kinetic equation is introduced to describe the collective modes and Fermi surface instabilities in a fluid of particles whose fermionic degrees of freedom obey the Majorana reality condition. …
Majorana modes and fractional fermions are two types of edge zero modes appearing separately in topological superconductors and dimerized chains. Here we reveal how to harvest both types of edge modes simultaneously in an exotic chain. Such modes are naturally spin-charge separated, and are protected by the inversion and spin-parity symmetries. We construct a lattice model to illustrate the nature of these edge modes, utilizing fermionic functional renormalization group, mean-field theory and bosonization. We also elucidate that the four-fold degenerate ground states with edge-spinons in the H…
We revisit the fermionic parton approach to S = 1/2 quantum spin liquids with SU(2) spin rotation symmetry, and the associated projective symmetry group (PSG) classification. We point out that the existing PSG classification is incomplete; upon completing it, we find spin liquid states with S=1 and S=0 Majorana fermion excitations coupled to a deconfined Z2 gauge field. The crucial observation leading us to this result is that, like space group and time reversal symmetries, spin rotations can act projectively on the fermionic partons; that is, a spin rotation may be realized by simultaneous SU…
A new twist on the Majorana surface code: Bosonic and fermionic defects for fault-tolerant quantum computation
本文提出马约拉纳表面码中扭曲缺陷的统一框架，证明其可编码两倍拓扑保护信息，并实现通用计算，为容错量子计算开辟新路径。
In the Majorana representation of a spin 1/2 we find an identity which relates spin-spin correlators to one-particle fermionic correlators. This should be contrasted with the straightforward approach in which two-particle (four-fermion) correlators need to be calculated. We discuss applications to the analysis of the dynamics of a spin coupled to a dissipative environment and of a quantum detector performing a continuous measurement of a qubit's state.
Bosonization and Kramers-Wannier dualities in general dimensions
通过规范费米子宇称，将Majorana链与横场Ising模型的Kramers-Wannier对偶推广到任意维度，建立了费米子与自旋系统之间的酉等价关系，并导出了高维非可逆KW自对偶性。
A Class of Exactly Solvable Hamiltonians for S=1/2 Quantum Magnets with Spinless Fermionic Excitations in Higher Dimensions
本文提出一类高维S=1/2量子磁体的精确可解哈密顿量，通过非微扰利用键依赖耦合稳定分数化无自旋费米子自旋子和Majorana费米子激发。
Parafermions are emergent excitations which generalize Majorana fermions and are potentially relevant to topological quantum computation. Using the concept of Fock parafermions, we present a mapping between lattice $\mathbb{Z}_4$ parafermions and lattice spin-$1/2$ fermions which preserves the locality of operators with $\mathbb{Z}_4$ symmetry. Based on this mapping, we construct an exactly solvable, local, and interacting one-dimensional fermionic Hamiltonian which hosts zero-energy modes obeying parafermionic algebra. We numerically show that this parafermionic phase remains stable in a wide…
We study a system of interacting spinless fermions in one dimension which, in the absence of interactions, reduces to the Kitaev chain [A. Yu Kitaev, Phys.-Usp. \textbf{44}, 131 (2001)]. In the non-interacting case, a signal of topological order appears as zero-energy modes localized near the edges. We show that the exact ground states can be obtained analytically even in the presence of nearest-neighbor repulsive interactions when the on-site (chemical) potential is tuned to a particular function of the other parameters. As with the non-interacting case, the obtained ground states are two-fol…
Ground states of quadratic Hamiltonians for fermionic systems can be characterized in terms of orthogonal complex structures. The standard way in which such Hamiltonians are diagonalized makes use of a certain "doubling" of the Hilbert space. In this work we show that this redundancy in the Hilbert space can be completely lifted if the relevant orthogonal structure is taken into account. Such an approach allows for a treatment of Majorana fermions which is both physically and mathematically transparent. Furthermore, an explicit connection between orthogonal complex structures and the topologic…
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