Spin Space Research Articles

A continuum of compass spin models on the honeycomb lattice

Quantum spin models with spatially dependent interactions, known as compass models, play an important role in the study of frustrated quantum magnetism. One example is the Kitaev model on the honeycomb lattice with spin-liquid (SL) ground states and anyonic excitations. Another example is the geometrically frustrated quantum 120° model on the same lattice whose ground state has not been unambiguously established. To generalize the Kitaev model beyond the exactly solvable limit and connect it with other compass models, we propose a new model, dubbed ‘the tripod model’, which contains a continuum of compass-type models. It smoothly interpolates the Ising model, the Kitaev model, and the quantum 120° model by tuning a single parameter , the angle between the three legs of a tripod in the spin space. Hence it not only unifies three paradigmatic spin models, but also enables the study of their quantum phase transitions. We obtain the phase diagram of the tripod model numerically by tensor networks in the thermodynamic limit. We show that the ground state of the quantum 120° model has long-range dimer order. Moreover, we find an extended spin-disordered (SL) phase between the dimer phase and an antiferromagnetic phase. The unification and solution of a continuum of frustrated spin models as outline here may be useful to exploring new domains of other quantum spin or orbital models.

Twistor spinors and quasi-twistor spinors

The author studies the properties and applications of quasi-Killing spinors and quasi-twistor spinors and obtains some vanishing theorems. In particular, the author classifies all the types of quasi-twistor spinors on closed Riemannian spin manifolds. As a consequence, it is known that on a locally decomposable closed spin manifold with nonzero Ricci curvature, the space of twistor spinors is trivial. Some integrability condition for twistor spinors is also obtained.

The Effect of Planar Magnetic Inhomogeneities on the Critical Temperature of Ferromagnet–Superconductor Systems

We study superconducting systems with the inhomogeneous effective exchange field background. A model of magnetic superconductor which takes into account the collectivized electrons interaction with the inhomogeneous effective exchange field is used. With local unitary rotation in spinor space we rewrite the Hamiltonian in a new basis where this interaction is diagonal. The problem is reduced to the one with a uniform exchange field but the effective tensor field appears. This method allows us to simplify the Gor’kov, Eilenberger, and Usadel equations in many symmetric cases. We calculate the critical temperature of the superconductor/ferromagnet proximity system in the dirty limit where the ferromagnet has periodic domain structure with planar domain walls.

Fischer decomposition for ‐monogenics in quaternionic Clifford analysis

Spaces of spinor‐valued homogeneous polynomials, and in particular spaces of spinor‐valued spherical harmonics, are decomposed in terms of irreducible representations of the symplectic group Sp(p). These Fischer decompositions involve spaces of homogeneous, so‐called ‐monogenic polynomials, the Lie super algebra being the Howe dual partner to the symplectic group Sp(p). In order to obtain Sp(p)‐irreducibility, this new concept of ‐monogenicity has to be introduced as a refinement of quaternionic monogenicity; it is defined by means of the four quaternionic Dirac operators, a scalar Euler operator underlying the notion of symplectic harmonicity and a multiplicative Clifford algebra operator P underlying the decomposition of spinor space into symplectic cells. These operators and P, and their Hermitian conjugates, arise naturally when constructing the Howe dual pair , the action of which will make the Fischer decomposition multiplicity free. Copyright © 2016 John Wiley & Sons, Ltd.

Magnetic moments of JP = $ {\frac{{3}}{{2}}}$ + decuplet baryons using the statistical model

A suitable wave function for baryon decuplet is framed with inclusion of sea containing quark- gluon Fock states. Relevant operator formalism is applied to calculate magnetic moments of JP = 3 2 + baryon decuplet. Statistical model assumes decomposition of baryonic state in various quark-gluon Fock states such as jqqqijgi; jqqqijggi; jqqqijgggi with possibility gluon emitting qq pairs condensates due to the subprocesses like g , qq; g , gg and g , qg where qq = uu; dd; ss. Statistical approach and detailed balance principle in combination is used to find the relative probabilities of these Fock states in avor, spin and color space. The total number of partons (sea) in this formalism are restricted to three gluons due to limited free energy of gluon and suppressed number of strange quark-antiquark pairs. The combined approach is used to calculate the magnetic moments, importance of strangeness in the sea (scalar, vector and tensor). Our approach has confirmed the scalar-tensor sea dominancy over vector sea. Various modifications in the model are used to check the validity of statistical approach. The results are matched with theoretical data available. Good consistency with the experimental data have been achieved for Delta++, Delta+ and Omega-.

Krein Spectral Triples and the Fermionic Action

Motivated by the space of spinors on a Lorentzian manifold, we define Krein spectral triples, which generalise spectral triples from Hilbert spaces to Krein spaces. This Krein space approach allows for an improved formulation of the fermionic action for almost-commutative manifolds. We show by explicit calculation that this action functional recovers the correct Lagrangians for the cases of electrodynamics, the electro-weak theory, and the Standard Model. The description of these examples does not require a real structure, unless one includes Majorana masses, in which case the internal spaces also exhibit a Krein space structure.

The quantum consistency of the 12-dimensional theory

By considering the 12-dimensional superalgebra, inferences are drawn about the finiteness of the 12-dimensional theory unifying the superstring models. The dimensional reduction of the nonsupersymmetric theory in four dimensions to a supersymmetric action in three dimensions is established for the bosonic sector. It is found to be the quotient by [Formula: see text] of the integration over the fiber coordinate of a theory with [Formula: see text] supersymmetry. Consequently, a flow on the moduli space of Spin(7) manifolds from a [Formula: see text] structure with [Formula: see text] supersymmetry yielding a phenomelogically realistic particle spectrum to a [Formula: see text] holonomy manifold compatible with supersymmetry in three dimensions and a nonsupersymmetric action in four dimensions, solving the quantum cosmological constant problem, is proven to exist. The projection of the representations of the [Formula: see text] superalgebra of the 12-dimensional theory to four dimensions include nonperturbative string solitons that are more stable because the dynamics is described by supersymmetric theory with a higher degree of finiteness.

The accurate wavefunction of the active space of the rhenium dimer resolved using the ab initio Brueckner coupled-cluster method

Rhenium is a unique metal in the 5d-series of transition metals having the highest boiling point in the periodic table. It is also known to exist in poly-coordinated states with other rhenium atoms. Based on the existence of strikingly unusual states of elements in astrophysical bodies of nebulae, interstellar debris, exoplanets and other part of the universe, a set of ab initio calculations of the rhenium dimer has been conducted to provide detailed description of its molecular properties that are applicable to the astrochemical research. Ab initio calculations and NBO analysis revealed that rhenium forms quintuple bond in its diatomic state and that it displays preferred state of triplet configuration with high-lying electrons. Calculations also revealed that the two states of rhenium dimer vary in their bonding nature. The singlet spin space is composed of five single bonds, while the triplet state comprises four bonds and two additional lone pairs. Interestingly, while these two states vary in subdivision of electrons at the highest d-level, they share the same frequencies while having different zero-point energies. The calculations reveal intrinsic synergy between the atoms composed of natural bond orbitals, the bonding pattern and the thermochemical properties of Re2, all features being of significant importance to physical and chemical sciences.

Existence of featureless paramagnets on the square and the honeycomb lattices in 2+1 dimensions

The peculiar features of quantum magnetism sometimes forbid the existence of gapped `featureless' paramagnets which are fully symmetric and unfractionalized. The Lieb-Schultz-Mattis theorem is an example of such a constraint, but it is not known what the most general restriction might be. We focus on the existence of featureless paramagnets on the spin-1 square lattice and the spin-1 and spin-1/2 honeycomb lattice with spin rotation and space group symmetries in 2+1D. Although featureless paramagnet phases are not ruled out by any existing theorem, field theoretic arguments disfavor their existence. Nevertheless, by generalizing the construction of Affleck, Kennedy, Lieb and Tasaki to a class we call `slave-spin' states, we propose featureless wave functions for these models. The featureless-ness of the spin-1 slave-spin states on the square and honeycomb lattice are verified both analytically and numerically, but the status of the spin-1/2 honeycomb state remains unclear.

Quantum Magnetic Oscillations of the Surface Tension at a Metal-Insulator Interface

Any metal–insulator transition (MI transition) in a crystalline material must be a transition from a situation in which electronic bands overlap to a situation when they do not (Mott, Metal–insulator, 2nd edn. Taylor@Francis, London, 1990). For this case the self-consistent equations for the two-band conductor are formulated (cf. Dubovskii, JETP Lett. 99(1):22–26, 2014). The description of the MI phase transition is based on two order parameters. The first one is the material density distribution at the MI boundary \(\rho ({\vec {r}})\). The second one is a four-component complex vector in spin space \(\Upsilon ({\vec {r}})\). The value \(\Upsilon ({\vec {r}})\) determines the electron density in the metallic or semimetallic phase in the presence of an external magnetic field. Two different components of the vector describe possible spin states of electrons and holes inserted in the external magnetic field. The solution gives a singular behavior of the surface tension at the MI interface in the vicinity of the MI phase transition. At low temperature quantum oscillations of the surface tension in the magnetic field take place.

There Is a Way to Comprise Half-Integer Eigenvalues for Photon Spin

In this article, an attempt based on Spin Topological Space, STS, to give areasonable detailed account of the cause of photonic fermionization phenomena of light photon is made. STS is an unconventional spin space in quantum mechanics, which can be used to account for where the unconventional half-integer spin eigenvalues phenomenon of light photon comes from. We suggest to dectect the possible existence of photonic one-third-spinization phenomenon of light photon, by using three beams of light photon in interference experiment.

Phase Diagram and Quantum Order by Disorder in the KitaevK1−K2Honeycomb Magnet

We show that the topological Kitaev spin liquid on the honeycomb lattice is extremely fragile against the second-neighbor Kitaev coupling $K_2$, which has recently been shown to be the dominant perturbation away from the nearest-neighbor model in iridate Na$_2$IrO$_3$, and may also play a role in $\alpha$-RuCl$_3$ and Li$_2$IrO$_3$. This coupling naturally explains the zigzag ordering (without introducing unrealistically large longer-range Heisenberg exchange terms) and the special entanglement between real and spin space observed recently in Na$_2$IrO$_3$. Moreover, the minimal $K_1$-$K_2$ model that we present here holds the unique property that the classical and quantum phase diagrams and their respective order-by-disorder mechanisms are qualitatively different due to the fundamentally different symmetries of the classical and quantum counterparts.

Simultaneous emergence of curved space–time and quantum mechanics

It is shown in this paper that the geometrically structureless space–time manifold is converted instantaneously to a curved, a Riemannian, or may be a Finslerian space–time with an associated Riemannian space–time, on the appearance of quantum Weyl spinors dependent only on time in a background flat manifold and having the symplectic property in the abstract space of spinors. The scenario depicts simultaneous emergence of gravity in accord with general relativity and quantum mechanics. The emergent gravity leads to the generalized uncertainty principle, which in turn ushers in discrete space–time. The emerged space–time is specified here as to be Finslerian and the field equation in that space–time has been obtained from the classical one due to the arising quantized space and time. From this field equation we find the quantum field equation for highly massive (of the Planck order) spinors in the associated Riemannian space of the Finsler space, which is in fact, the background homogeneous and isotropic Friedmann–Robertson–Walker space–time of the universe. These highly massive spinors provide the mass distribution complying with the Einstein equivalence principle. All these occurred in the indivisible minimum time considered as zero time or spontaneity.

Asymptotic expressions for the hyperfine populations in the ground state of spin-1 condensates against a magnetic field

Based on perturbation theory up to second order, analytical asymptotic expressions for the variation of the population of hyperfine component $\ensuremath{\mu}=0$ particles in the ground state of spin-1 condensates against a magnetic field $B$ have been derived. The ranges of $B$ in which the asymptotic expressions are applicable have been clarified via a comparison of the numerical results from the analytical expressions and from a diagonalization of the Hamiltonian in a complete spin space. It was found that, for $^{87}\mathrm{Rb}$, the two analytical expressions, one for a weak field and the other one for a strong field, together cover the whole range of $B$ from zero to infinity. For Na, the analytical expressions are valid only if $B$ is very weak or sufficiently strong.

Rare-Earth Triangular Lattice Spin Liquid: A Single-Crystal Study of YbMgGaO4.

YbMgGaO4, a structurally perfect two-dimensional triangular lattice with an odd number of electrons per unit cell and spin-orbit entangled effective spin-1/2 local moments for the Yb(3+) ions, is likely to experimentally realize the quantum spin liquid ground state. We report the first experimental characterization of single-crystal YbMgGaO4 samples. Because of the spin-orbit entanglement, the interaction between the neighboring Yb(3+) moments depends on the bond orientations and is highly anisotropic in the spin space. We carry out thermodynamic and the electron spin resonance measurements to confirm the anisotropic nature of the spin interaction as well as to quantitatively determine the couplings. Our result is a first step towards the theoretical understanding of the possible quantum spin liquid ground state in this system and sheds new light on the search for quantum spin liquids in strong spin-orbit coupled insulators.

Expansions for the Dirac Operator and Related Operators in Super Spinor Space

In this paper, we study expansions for the Dirac operator D, the modified Dirac operator \({D-\lambda,}\) and the polynomial Dirac operator P(D) in super spinor space. These expansions are a meaningful generalization of the classical Almansi expansion in polyharmonic functions theory. As an application of the expansions, the generalized Riquier problem in super spinor space is investigated.

Classical lattice spin models involving singular interactions isotropic in spin space.

We address here a few classical lattice spin models, involving n-component unit vectors (n=2,3), associated with a D-dimensional lattice Z(D),D=1,2, and interacting via a pair potential restricted to nearest neighbors and being isotropic in spin space, i.e., defined by a function of the scalar product between the interacting spins. When the potential involves a continuous function of the scalar product, the Mermin-Wagner theorem and its generalizations exclude orientational order at all finite temperatures in the thermodynamic limit, and exclude phase transitions at finite temperatures when D=1; on the other hand, we have considered here some comparatively simple functions of the scalar product which are bounded from below, diverge to +∞ for certain mutual orientations, and are continuous almost everywhere with integrable singularities. Exact solutions are presented for D=1, showing an absence of phase transitions and an absence of orientational order at all finite temperatures in the thermodynamic limit; for D=2, and in the absence of more stringent mathematical results, extensive simulations carried out on some of them point to the absence of orientational order at all finite temperatures and suggest the existence of a Berezinskiĭ-Kosterlitz-Thouless transition.

Can a T2 hyperintense rim sign differentiate uterine leiomyomas from other solid adnexal masses?

To investigate the incidence of high T2 signal rims surrounding leiomyomas, evaluate if a particular T2-weighted sequence is more effective in depicting this rim, and determine if this sign is useful in differentiating pedunculated leiomyomas from other solid adnexal masses. In this retrospective study, two radiologists evaluated 233 T2 dark pelvic masses (223 uterine leiomyomas and 10 ovarian fibromas) in 60 women (mean age 47) on Magnetic resonance imaging for the presence of a high signal rim. Three different T2-weighted sequences were reviewed independently for uterine leiomyomas: half-Fourier acquisition single-shot turbo spin echo (HASTE), SPACE, and T2 with fat saturation (T2 FS). Only T2 FS images were available for 10 fibromas. A consensus review was conducted for discrepant cases. Statistical analyses were performed using Fisher's exact test, kappa test, and ANOVA RESULTS: For 223 uterine leiomyomas, 23% (95% CI 17.8-28.9%) demonstrated a high T2 signal rim sign on T2 FS compared with 4.9% (95% CI 2.6-8.9%) for HASTE and 6.7% (95% CI 3.9-11.1%) for SPACE. The difference between the number of positive rims on T2 FS relative-HASTE and SPACE was statistically significant (p < 0.001). For ovarian fibromas, 40% (95% CI 16.9-68.8%) were classified positive for a rim sign. A high T2 signal rim sign was present for up to 23% of uterine leiomyomas and the T2 FS sequence detected this rim sign most frequently. Up to 40% of ovarian fibromas can also have a T2 rim sign and, therefore, a solid adnexal mass with a T2 rim sign cannot be assumed to represent a pedunculated leiomyoma.

Clifford Algebra is the Natural Framework for Root Systems and Coxeter Groups. Group Theory: Coxeter, Conformal and Modular Groups

In this paper, we make the case that Clifford algebra is the natural framework for root systems and reflection groups, as well as related groups such as the conformal and modular groups: The metric that exists on these spaces can always be used to construct the corresponding Clifford algebra. Via the Cartan-Dieudonn\'e theorem all the transformations of interest can be written as products of reflections and thus via `sandwiching' with Clifford algebra multivectors. These multivector groups can be used to perform concrete calculations in different groups, e.g. the various types of polyhedral groups, and we treat the example of the tetrahedral group $A_3$ in detail. As an aside, this gives a constructive result that induces from every 3D root system a root system in dimension four, which hinges on the facts that the group of spinors provides a double cover of the rotations, the space of 3D spinors has a 4D euclidean inner product, and with respect to this inner product the group of spinors can be shown to be closed under reflections. In particular the 4D root systems/Coxeter groups induced in this way are precisely the exceptional ones, with the 3D spinorial point of view also explaining their unusual automorphism groups. This construction simplifies Arnold's trinities and puts the McKay correspondence into a wider framework. We finally discuss extending the conformal geometric algebra approach to the 2D conformal and modular groups, which could have interesting novel applications in conformal field theory, string theory and modular form theory.

Conformal symmetries of the super Dirac operator

In this paper, the Dirac operator, acting on super functions with values in super spinor space, is defined along the lines of the construction of generalized Cauchy–Riemann operators by Stein and Weiss. The introduction of the superalgebra of symmetries \mathfrak{osp}(m|2n) is a new and essential feature in this approach. This algebra of symmetries is extended to the algebra of conformal symmetries \mathfrak{osp}(m+1,1|2n) . The kernel of the Dirac operator is studied as a representation of both algebras. The construction also gives an explicit realization of the Howe dual pair \mathfrak{osp}(1|2)\times\mathfrak{osp}(m|2n)\subset \mathfrak{osp}(m+4n|2m+2n) . Finally, the super Dirac operator gives insight into the open problem of classifying invariant first order differential operators in super parabolic geometries.