Spinor Representation Research Articles

Hierarchy of correlations for the Ising model in the Majorana representation

We study the quantum Ising model in D dimensions with the equation of motion technique, in combination with the Majorana representation for spins. The decoupling scheme used for the Green's functions is based on the hierarchy of correlations in position space. To lowest order this method reproduces the well known mean field phase diagram and critical exponents. When correlations between spins are included, we show how the appearance of thermal fluctuations and magnons strongly affect the physical properties. We demonstrate that in 1D and for B=0, thermal fluctuations completely destroy the ordered phase, and that near the quantum critical point, the quantum model displays different behavior than its classical counterpart. We discuss the connection with the Dyson's equation formalism and the explicit form of the self-energies.

Spinorial representation of submanifolds in Riemannian space forms

In this paper we give a spinorial representation of submanifolds of any dimension and codimension into Riemannian space forms in terms of the existence of so called generalized Killing spinors. We then discuss several applications, among them a new and concise proof of the fundamental theorem of submanifold theory. We also recover results of T. Friedrich, B. Morel and the authors in dimension 2 and 3.

The (restricted) Inomata-McKinley spinor representation and the underlying topology

The so-called Inomata-McKinley spinors are a particular solution of the non-linear Heisenberg equation. In fact, free linear massive (or massless) Dirac fields are well known to be represented as a combination of Inomata-McKinley spinors. More recently, a subclass of Inomata-McKinley spinors was used to describe neutrino physics. In this paper we show that Dirac spinors undergoing this restricted Inomata-McKinley decomposition are necessarily of the first type, according to the Lounesto classification. Moreover, we also show that these type-one subclass spinors have not an exotic counterpart. Finally, implications of these results are discussed, regarding the understanding of the spacetime background topology.

Subsectors, Dynkin diagrams and new generalised geometries

We examine how generalised geometries can be associated with a labelled Dynkin diagram built around a gravity line. We present a series of new generalised geometries based on the groups Spin(d, d) × ℝ+ for which the generalised tangent space transforms in a spinor representation of the group. In low dimensions these all appear in subsectors of maximal supergravity theories. The case d = 8 provides a geometry for eight-dimensional backgrounds of M theory with only seven-form flux, which have not been included in any previous geometric construction. This geometry is also one of a series of “half-exceptional” geometries, which “geometrise” a six-form gauge field. In the appendix, we consider exam-ples of other algebras appearing in gravitational theories and give a method to derive the Dynkin labels for the “section condition” in general. We argue that generalised geometry can describe restrictions and subsectors of many gravitational theories.

Single-spin entanglement

We show that the operators and the quadrupole and Zeeman Hamiltonians for a spin (3/2) can be represented in terms for a system of two coupling fictitious spins (1/2) using the Kronecker product of Pauli matrices. Particularly, the quadrupole Hamiltonian which describes the interaction of the nuclear quadrupole moment with an electric field gradient is represented as the Hamiltonian of Ising model in a transverse selective magnetic field. The Zeeman Hamiltonian, which describes interaction of the nuclear spin with the external magnetic field, can be considered as the Hamiltonian of the Heisenberg model in a selective magnetic field. The total Hamiltonian can be interpreted as the Hamiltonian of 3D Heisenberg model in an inhomogeneous magnetic field applied along the x-axis. The representation of a single spin (3/2) as two-spin (1/2) system allows us to study entanglement in the spin system. One of the features of the fictitious spin system is that, in both the pure and the mixed states, the concurrence tends to 0.5 with increase of applied magnetic field. The representation of a spin (3/2) as a system of two coupling fictitious spins (1/2) and possibility of formation of the entangled states in this system open a way to the application of a single spin (3/2) in quantum computation.

On the spinor representation

A systematic study of the spinor representation by means of the fermionic physical space is accomplished and implemented. The spinor representation space is shown to be constrained by the Fierz–Pauli–Kofink identities among the spinor bilinear covariants. A robust geometric and topological structure can be manifested from the spinor space, wherein the first and second homotopy groups play prominent roles on the underlying physical properties, associated to fermionic fields. The mapping that changes spinor fields classes is then exemplified, in an Einstein–Dirac system that provides the spacetime generated by a fermion.

A quantum fermion realisation of the finite dimensional spinor representation of Uq(osp(1|2n))

The quantum supergroup Uq(osp(1|2n)) admits a finite dimensional spinor representation, which does not have a classical limit. We construct a realisation of this representation on the Fock space of q-fermions. We also generalise the construction to the infinite dimensional spinor representations of Uq(osp(2m+1|2n)) for m,n≥1 by using both quantum bosons and fermions. This leads to a new realisation different from those obtained before by other researchers.

Quantum to classical transition in the ground state of a spin-S quantum antiferromagnet

We study a frustrated spin-$S$ staggered-dimer Heisenberg model on square lattice by using the bond-operator representation for quantum spins, and investigate the emergence of classical magnetic order from the quantum mechanical (staggered-dimer singlet) ground state for increasing $S$. Using triplon analysis, we find the critical couplings for this quantum phase transition to scale as $1/S(S+1)$. We extend the triplon analysis to include the effect of quintet dimer-states, which proves to be essential for establishing the classical order (N\'eel or collinear in the present study) for large $S$, both in the purely Heisenberg case and also in the model with single-ion anisotropy.

Wigner\u2019s infinite spin representations and inert matter

Positive energy ray representations of the Poincaré group are naturally subdivided into three classes according to their mass and spin content: m > 0, m=0 finite helicity and m=0 infinite spin. For a long time the localization properties of the massless infinite spin class remained unknown, until it became clear that such matter does not permit compact spacetime localization and its generating covariant fields are localized on semi-infinite space-like strings. Using a new perturbation theory for higher spin fields we present arguments which support the idea that infinite spin matter cannot interact with normal matter and we formulate conditions under which this also could happen for finite spin s>1 fields. This raises the question of a possible connection between inert matter and dark matter.

New Hamiltonians for loop quantum cosmology with arbitrary spin representations

In loop quantum cosmology, one has to make a choice of SU(2) irreducible representation in which to compute holonomies and regularize the curvature of the connection. The systematic choice made in the literature is to work in the fundamental representation, and very little is known about the physics associated with higher spin labels. This constitutes an ambiguity of which the understanding, we believe, is fundamental for connecting loop quantum cosmology to full theories of quantum gravity like loop quantum gravity, its spin foam formulation, or cosmological group field theory. We take a step in this direction by providing here a new closed formula for the Hamiltonian of flat Friedmann-Lema\^{\i}tre-Robertson-Walker models regularized in a representation of arbitrary spin. This expression is furthermore polynomial in the basic variables which correspond to well-defined operators in the quantum theory, takes into account the so-called inverse-volume corrections, and treats in a unified way two different regularization schemes for the curvature. After studying the effective classical dynamics corresponding to single and multiple-spin Hamiltonians, we study the behavior of the critical density when the number of representations is increased and the stability of the difference equations in the quantum theory.

The simulation of non-Abelian statistics of Majorana fermions in Ising chain with Z2 symmetry

In this paper, we numerically study the non-Abelian statistics of the zero-energy Majorana fermions at the end of Majorana chain and show its application to quantum computing by mapping it to a spin model with special symmetry. In particular, by using transverse-field Ising model with Z2 symmetry, we verify the nontrivial non-Abelian statistics of Majorana fermions. Numerical evidence and comparison in both Majorana representation and spin representation are presented. The degenerate ground states of a symmetry protected spin chain therefore provide a promising platform for topological quantum computation.

Infinite rank spinor and oscillator representations

We develop a functorial theory of spinor and oscillator representations parallel to the theory of Schur functors for general linear groups. This continues our work on developing orthogonal and symplectic analogues of Schur functors. As such, there are a few main points in common. We define a category of representations of what might be thought of as the infinite rank pin and metaplectic groups, and give three models of this category in terms of: multilinear algebra, diagram categories, and twisted Lie algebras. We also define specialization functors to the finite rank groups and calculate the derived functors.

Relation between discrete Frenet frames and the bi-Hamiltonian structure of the discrete nonlinear Schrödinger equation

The discrete Frenet equation entails a local framing of a discrete, piecewise linear polygonal chain in terms of its bond and torsion angles. In particular, the tangent vector of a segment is akin the classical O(3) spin variable. Thus there is a relation to the lattice Heisenberg model, that can be used to model physical properties of the chain. On the other hand, the Heisenberg model is closely related to the discrete nonlinear Schr\"odinger (DNLS) equation. Here we apply these interrelations to develop a perspective on discrete chains dynamics: We employ the properties of a discrete chain in terms of a spinorial representation of the discrete Frenet equation, to introduce a bi-hamiltonian structure for the discrete nonlinear Schr\"odinger equation (DNLSE), which we then use to produce integrable chain dynamics.

Extension of the correspondence to the infinite-dimensional chiral spinors and self dual tensors

The spinor representations of the orthosymplectic Lie superalgebras are considered and constructed. These are infinite-dimensional irreducible representations, of which the superdimension coincides with the dimension of the spinor representation of . Next, we consider the self dual tensor representations of and their generalizations: these are also infinite-dimensional and correspond to the highest irreducible component of the pth power of the spinor representation. We determine the character of these representations, and deduce a superdimension formula. From this, it follows that also for these representations the correspondence holds.

On Spinor Representations in Galilean and Pseudo-Galilean Geometry

On Spinor Representations in Galilean and Pseudo-Galilean Geometry

The coprime quantum chain

In this paper we introduce and study the coprime quantum chain, i.e. a strongly correlated quantum system defined in terms of the integer eigenvalues ni of the occupation number operators at each site of a chain of length M. The ni’s take value in the interval [2,q] and may be regarded as Sz eigenvalues in the spin representation j = (q − 2)/2. The distinctive interaction of the model is based on the coprimality matrix : for the ferromagnetic case, this matrix assigns lower energy to configurations where occupation numbers ni and ni+1 of neighbouring sites share a common divisor, while for the anti-ferromagnetic case it assigns a lower energy to configurations where ni and ni+1 are coprime. The coprime chain, both in the ferro and anti-ferromagnetic cases, may present an exponential number of ground states whose values can be exactly computed by means of graph theoretical tools. In the ferromagnetic case there are generally also frustration phenomena. A fine tuning of local operators may lift the exponential ground state degeneracy and, according to which operators are switched on, the system may be driven into different classes of universality, among which the Ising or Potts universality class. The paper also contains an appendix by Don Zagier on the exact eigenvalues and eigenvectors of the coprimality matrix in the limit .

Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons, and Mirror Symmetry

We address the issue of why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two forms and four forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds. We explore how the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we show that the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian Yang-Mills equations on the Calabi-Yau manifold. Therefore, the mirror symmetry of Calabi-Yau manifolds can be recast as the mirror pair of Hermitian Yang-Mills instantons. We discuss the mirror symmetry from the gauge theory perspective.

Spinor representation of Maxwell’s equations

Spinors are more special objects than tensor. Therefore possess more properties than the more generic objects such as tensors. Thus, the group of Lorentz two-spinors is the covering group of the Lorentz group. Since the Lorentz group is a symmetry group of Maxwell’s equations, it is assumed to reasonable to use when writing the Maxwell equations Lorentz two-spinors and not tensors. We describe in detail the representation of the Maxwell’s equations in the form of Lorentz two-spinors. This representation of Maxwell’s equations can be of considerable theoretical interest.

Spinorial representation of submanifolds in metric Lie groups

In this paper we give a spinorial representation of submanifolds of any dimension and codimension into Lie groups equipped with left invariant metrics. As applications, we get a spinorial proof of the Fundamental Theorem for submanifolds into Lie groups, we recover previously known representations of submanifolds in $\mathbb{R}^n$ and in the 3-dimensional Lie groups $S^3$ and $E(\kappa,\tau),$ and we get a new spinorial representation for surfaces in the 3-dimensional semi-direct products: this achieves the spinorial representations of surfaces in the 3-dimensional homogeneous spaces. We finally indicate how to recover a Weierstrass-type representation for CMC-surfaces in 3-dimensional metric Lie groups recently given by Meeks, Mira, Perez and Ros.

Realization of the Haldane-Kane-Mele Model in a System of Localized Spins.

We study a spin Hamiltonian for spin-orbit-coupled ferromagnets on the honeycomb lattice. At sufficiently low temperatures supporting the ordered phase, the effective Hamiltonian for magnons, the quanta of spin-wave excitations, is shown to be equivalent to the Haldane model for electrons, which indicates the nontrivial topology of the band and the existence of the associated edge state. At high temperatures comparable to the ferromagnetic-exchange strength, we take the Schwinger-boson representation of spins, in which the mean-field spinon band forms a bosonic counterpart of the Kane-Mele model. The nontrivial geometry of the spinon band can be inferred by detecting the spin Nernst effect. A feasible experimental realization of the spin Hamiltonian is proposed.