Universal spin-triplet superconducting correlations of Majorana fermions

Abstract We use the SYK family of models with N Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such “shortcuts” through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at O($$ \sqrt{N} $$ N ), and we find an explicit operator which “fast-forwards” the free N-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by O(poly(N)), and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times O(eN), after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.

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Chaos on the hypercube

Journal of High Energy Physics ◽

10.1007/jhep11(2020)154 ◽

2020 ◽

Vol 2020 (11) ◽

Author(s):

Yiyang Jia ◽

Jacobus J. M. Verbaarschot

Keyword(s):

Magnetic Flux ◽

Spectral Density ◽

Hermite Polynomials ◽

Weighted Sums ◽

Majorana Fermions ◽

Chord Diagrams ◽

Magnetic Inversion ◽

Constant Magnitude ◽

Kitaev Model ◽

Spectral Statistics

Abstract We analyze the spectral properties of a d-dimensional HyperCubic (HC) lattice model originally introduced by Parisi. The U(1) gauge links of this model give rise to a magnetic flux of constant magnitude ϕ but random orientation through the faces of the hypercube. The HC model, which also can be written as a model of 2d interacting Majorana fermions, has a spectral flow that is reminiscent of Maldacena-Qi (MQ) model, and its spectrum at ϕ = 0, actually coincides with the coupling term of the MQ model. As was already shown by Parisi, at leading order in 1/d, the spectral density of this model is given by the density function of the Q-Hermite polynomials, which is also the spectral density of the double-scaled Sachdev-Ye-Kitaev model. Parisi demonstrated this by mapping the moments of the HC model to Q-weighted sums on chord diagrams. We point out that the subleading moments of the HC model can also be mapped to weighted sums on chord diagrams, in a manner that descends from the leading moments. The HC model has a magnetic inversion symmetry that depends on both the magnitude and the orientation of the magnetic flux through the faces of the hypercube. The spectrum for fixed quantum number of this symmetry exhibits a transition from regular spectra at ϕ = 0 to chaotic spectra with spectral statistics given by the Gaussian Unitary Ensembles (GUE) for larger values of ϕ. For small magnetic flux, the ground state is gapped and is close to a Thermofield Double (TFD) state.

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From Majorana fermions to topological quantum computation

2020 7th International Forum on Electrical Engineering and Automation (IFEEA) ◽

10.1109/ifeea51475.2020.00166 ◽

2020 ◽

Author(s):

Cheng Zhang

Keyword(s):

Quantum Computation ◽

Majorana Fermions ◽

Topological Quantum ◽

Topological Quantum Computation

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Iterants, Majorana Fermions and the Majorana-Dirac Equation

Symmetry ◽

10.3390/sym13081373 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1373

Author(s):

Louis H. Kauffman

Keyword(s):

Dirac Equation ◽

Schrödinger Equation ◽

Schrodinger Equation ◽

Clifford Algebras ◽

Discrete Systems ◽

Complex Numbers ◽

Majorana Fermions ◽

Matrix Algebras ◽

Dirac Equations ◽

Points Of View

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

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