Total Spin Operator Research Articles

Scaling limit of the staggered six-vertex model with $U_q\big(\mathfrak{sl}(2)\big)$ invariant boundary conditions

We study the scaling limit of a statistical system, which is a special case of the integrable inhomogeneous six-vertex model. It possesses U_q\big(\mathfrak{sl}(2)\big)Uq(𝔰𝔩(2)) invariance due to the choice of open boundary conditions imposed. An interesting feature of the lattice theory is that the spectrum of scaling dimensions contains a continuous component. By applying the ODE/IQFT correspondence and the method of the Baxter QQ operator the corresponding density of states is obtained. In addition, the partition function appearing in the scaling limit of the lattice model is computed, which may be of interest for the study of nonrational CFTs in the presence of boundaries. As a side result of the research, a simple formula for the matrix elements of the QQ operator for the general, integrable, inhomogeneous six-vertex model was discovered, that has not yet appeared in the literature. It is valid for a certain one parameter family of diagonal open boundary conditions in the sector with the zz-projection of the total spin operator being equal to zero.

Dissipative time crystals with long-range Lindbladians

Dissipative time crystals can appear in spin systems, when the ${Z}_{2}$ symmetry of the Hamiltonian is broken by the environment, and the square of total spin operator ${S}^{2}$ is conserved. In this paper, we relax the latter condition and show that time-translation-symmetry-breaking collective oscillations persist, in the thermodynamic limit, even in the absence of spin symmetry. We engineer an ad hoc Lindbladian using power-law-decaying spin operators and show that time-translation-symmetry breaking appears when the decay exponent obeys $0<\ensuremath{\eta}\ensuremath{\le}1$. This model shows a surprisingly rich phase diagram, including the time-crystal phase as well as first-order, second-order, and continuous transitions of the fixed points. We study the phase diagram and the magnetization dynamics in the mean-field approximation. We prove that this approximation is quantitatively accurate, when $0<\ensuremath{\eta}<1$ and the thermodynamic limit is taken, because the system does not develop sizable quantum fluctuations, if the Gaussian approximation is considered.

Asymptotic optimality of twist-untwist protocols for Heisenberg scaling in atom-based sensing

Twist-untwist protocols for quantum metrology consist of a serial application of (1) unitary nonlinear dynamics (e.g., spin squeezing or Kerr nonlinearity), (2) parameterized dynamics $U(\ensuremath{\phi})$ (e.g., a collective rotation or phase space displacement), and (3) time reversed application of step 1. Such protocols are known to produce states that allow Heisenberg scaling for experimentally accessible estimators of $\ensuremath{\phi}$ even when the nonlinearities are applied for times much shorter than required to produce Schr\"odinger cat states. In this work, we prove that, asymptotically in the number of particles, twist-untwist protocols provide the lowest estimation error among quantum metrology protocols that utilize two calls to a weakly nonlinear evolution and a readout involving only first and second moments of a total spin operator $\stackrel{P\vec}{n}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{J}$. We consider the following physical settings: all-to-all interactions generated by one-axis twisting ${J}_{z}^{2}$ (e.g., interacting Bose gases), constant finite range spin-spin interactions of distinguishable or bosonic atoms (e.g., trapped ions or Rydberg atoms, or lattice bosons). In these settings, we further show that the optimal twist-untwist protocols asymptotically achieve 85% and 92% of the respective quantum Cram\'er-Rao bounds. We show that the error of a twist-untwist protocol can be decreased by a factor of $L$ without an increase in the noise of the spin measurement if the twist-untwist protocol can be noiselessly iterated as an $L$ layer quantum alternating operator ansatz.

Error suppression in adiabatic quantum computing with qubit ensembles

Incorporating protection against quantum errors into adiabatic quantum computing (AQC) is an important task due to the inevitable presence of decoherence. Here, we investigate an error-protected encoding of the AQC Hamiltonian, where qubit ensembles are used in place of qubits. Our Hamiltonian only involves total spin operators of the ensembles, offering a simpler route towards error-corrected quantum computing. Our scheme is particularly suited to neutral atomic gases where it is possible to realize large ensemble sizes and produce ensemble-ensemble entanglement. We identify a critical ensemble size Nc where the nature of the first excited state becomes a single particle perturbation of the ground state, and the gap energy is predictable by mean-field theory. For ensemble sizes larger than Nc, the ground state becomes protected due to the presence of logically equivalent states and the AQC performance improves with N, as long as the decoherence rate is sufficiently low.

Efficient formulation of full configuration interaction quantum Monte Carlo in a spin eigenbasis via the graphical unitary group approach.

We provide a spin-adapted formulation of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) algorithm, based on the Graphical Unitary Group Approach (GUGA), which enables the exploitation of SU(2) symmetry within this stochastic framework. Random excitation generation and matrix element calculation on the Shavitt graph of GUGA can be efficiently implemented via a biasing procedure on the branching diagram. The use of a spin-pure basis explicitly resolves the different spin-sectors and ensures that the stochastically sampled wavefunction is an eigenfunction of the total spin operator S^2. The method allows for the calculation of states with low or intermediate spin in systems dominated by Hund's first rule, which are otherwise generally inaccessible. Furthermore, in systems with small spin gaps, the new methodology enables much more rapid convergence with respect to walker number and simulation time. Some illustrative applications of the GUGA-FCIQMC method are provided: computation of the 2F - 4F spin gap of the cobalt atom in large basis sets, achieving chemical accuracy to experiment, and the Σg+1, Σg+3, Σg+5, and Σg+7 spin-gaps of the stretched N2 molecule, an archetypal strongly correlated system.

Random spin distributions and the diffusion equation

We show that the probability distribution corresponding to a fully random tracial state of a system of spin-S particles satisfies a diffusion-like equation. The diffusion coefficient turns out to be equal to S(S+1)/6, where S is the magnitude of the spin of each particle. We also present a bosonization scheme for the lowering and raising total spin operators.

Theoretical study on the magnetic moments formation in Ta-doped anatase TiO2

We present a theoretical study on Ti-vacancy induced ferromagnetism in Ta-doped anatase TiO2. Experimental study of Ti1−xTaxO2 thin film has shown that Ti-vacancies (assisted by Ta doping) induce the formation of localized magnetic moment around it, then, the observed ferromagnetism is caused by the alignment of localized magnetic moments through Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. In this study, we focus on the formation of the localized magnetic moments in this system. We hypothesize that on a unit cell, Ti-vacancy has caused four electrons from the surrounding oxygen atoms to become unpaired. These unpaired electrons then arrange themselves into a configuration with a non-zero net magnetic moment. To examine our hypothesis, we construct a Hamiltonian of the four unpaired electrons, incorporating the Coulomb intra- and inter-orbital interactions, in matrix form. Using a set of chosen parameter values, we diagonalize the Hamiltonian to get the eigenstates and eigenvalues, then, with the resulting eigenstates, we calculate the magnetic moment, μ, by obtaining the expectation value of the square of total spin operator. Our calculation results show that in the ground state, provided that the ratio of parameters satisfies some criterion, μ ≈ 4μB, corresponding to the four electron spins being almost perfectly aligned, can be achieved. Further, as long as we keep the Coulomb intra-orbital interaction between 0.5 and 1 eV, we find that μ ≈ 4μB is robust up to far above room temperature. Our results demonstrate that Ti vacancies in anatase TiO2 can form very stable localized magnetic moments.

Long-lived states with well-defined spins inspin−1/2homogeneous Bose gases

Many-body eigenfunctions of the total spin operator can be constructed from the spin and spatial wavefunctions with non-trivial permutation symmetries. Spin-dependent interactions can lead to relaxation of the spin eigenstates to the thermal equilibrium. A mechanism that stabilizes the many-body entangled states is proposed here. Surprisingly, in spite coupling with the chaotic motion of the spatial degrees of freedom, the spin relaxation can be suppressed by destructive quantum interference due to spherical vector and tensor terms of the spin-dependent interactions. Tuning the scattering lengths by the method of Feshbach resonances, readily available in cold atomic labs, can enhance the relaxation timescales by several orders of magnitude.

Approximate singly excited states from a two-component Hartree-Fock reference.

For many molecules, relaxing the spin symmetry constraint on the wave function results in the lowest energy mean-field solution. The two-component Hartree-Fock (2cHF) method relaxes all spin symmetry constraints, and the wave function is no longer an eigenfunction of the total spin, spin projection, or time-reversal symmetry operators. For ground state energies, 2cHF is a superior mean-field method for describing spin-frustrated molecules. For excited states, the utility of 2cHF is uncertain. Here, we implement the 2cHF extensions of two single-reference excited state methods, the two-component configuration interaction singles and time-dependent Hartree-Fock. We compare the results to the analogous methods based off of the unrestricted Hartree-Fock approximation, as well as the full configuration interaction for three small molecules with distinct 2cHF solutions, and discuss the nature of the 2cHF excited state solutions.

CheMPS2: Improved DMRG-SCF routine and correlation functions

CHEMPS2, our spin-adapted implementation of the density matrix renormalization group (DMRG) for ab initio quantum chemistry (Wouters et al., 2014), has several new features. A speed-up of the augmented Hessian Newton-Raphson DMRG self-consistent field (DMRG-SCF) routine is achieved with the direct inversion of the iterative subspace (DIIS). For extended molecules, the active space orbitals can be localized by maximizing the Edmiston-Ruedenberg cost function. These localized orbitals can be ordered according to the topology of the molecule by approximately minimizing the bandwidth of the exchange matrix with the Fiedler vector. The electronic structure can be analyzed by means of the two-orbital mutual information, spin, spin-flip, density, and singlet diradical correlation functions.

Symmetry and broken symmetry in molecular orbital description of unstable molecules IV: comparison between single- and multi-reference computational results for antiaromtic molecules

First principle calculations of the effective exchange integrals (J) in the Heisenberg model for diradical species are presented for both symmetry-adapted multi-reference (MR) and single-reference broken-symmetry (BS) methods. The Mukherjee-type state-specific MR coupled cluster singles and doubles (MkCCSD) method with several different reference orbitals including BS natural orbitals is used to calculate the singlet–triplet energy gaps (S–T energy gap or 2J) and diradical characters for the antiaromatic molecules [a cyclopropenyl anion (CPA), b cyclobutadiene (CBD), and c cyclopentadienyl cation (CPC)], the cyclobutadiene derivatives with polar substitutents [d aminocyclobutadiene (ACBD), e formylcyclobutadiene (FCBD), and f 1-amino-2-formyl-cyclobutadiene (AFCBD)] and finally the cyclobutadine derivatives with radical substitutents [g 1,2-bis(methylene)cyclobutadiene (1,2-BMCBD) and h 1,3-bis(methylene)cyclobutadiene (1,3-BMCBD)]. For the BS methods, the spin-unrestricted Hartree–Fock based CCSD (UHF-CCSD), the CCD with the spin-unrestricted Brueckner determinant (UBD), and BS density functional theory (UDFT) computations are performed. Comparison between MkCCSD and the UHF-CCSD results indicates that spin-contamination of UHF-CCSD solutions still remains. In comparison with UHF-CCSD, the UBD results show that spin-contamination involved in BS solutions is greatly suppressed. To eliminate the spin contamination, an approximate spin-projection (AP) scheme is applied to the BS solutions. The AP procedure with the use of the expectation value of the total-spin operator corresponding to UHF-CCSD and UBD results yields good agreement with the MkCCSD results. As for the AP correction of the UDFT methods, three different computational schemes for predicting the expectation value of the total-spin operator are examined. Systematic comparisons between these methods are presented for the S–T energy gaps (2J). Implications of the present computational results have been discussed in relation to the design of magnetic oligomers and polymers.

Berry's phases for arbitrary spins nonlinearly coupled to external fields: application to the entanglement of N > 2 non-correlated 1/2-spins

We derive the general formula giving Berry's phase for an arbitrary spin, having both magnetic-dipole and electric-quadrupole couplings with external time-dependent fields. We assume that the ‘effective’ electric and magnetic fields remain orthogonal during the quantum cycles. This mild restriction has many advantages. It provides simple symmetries leading to useful selection rules. Furthermore, the Hamiltonian parameter space coincides with the density matrix space for a spin S = 1. This implies a mathematical identity between Berry's phase and the Aharonov–Anandan phase, which is lost for S > 1. We have found that new physical features of Berry's phases emerge for integer spins ⩾2. We provide explicit numerical results of Berry's phases for S = 2, 3, 4. For any spin, one easily finds well-defined regions of the parameter space where instantaneous eigenstates remain widely separated. We have decided to stay within them and not deal with non-Abelian Berry's phases. We present a thorough and precise analysis of the non-adiabatic corrections with separate treatment for periodic and non-periodic Hamiltonian parameters. In both cases, the accuracy for satisfying the adiabatic approximation within a definite time interval is considerably improved if one chooses for the time derivatives of the parameters a time-dependence having a Blackman pulse shape. This has the effect of taming the non-adiabatic oscillation corrections which could be generated by a linear ramping. For realistic experimental conditions, the non-adibatic corrections can be kept below the 0.1% level. For a class of quantum cycles, involving as a sole periodic parameter the precession angle of the electric field around the magnetic field, the corrections odd upon the reversal of the associated rotation velocity can be canceled exactly if the quadrupole to dipole coupling is chosen appropriately (‘magic values’). The even ones are eliminated by taking the difference of the Berry phases generated by two ‘mirror’ cycles. We end by indicating a possible application of the theoretical tools developed in this paper. We propose a way to perform a holonomic entanglement of N non-correlated 1/2-spins by performing adiabatic cycles governed by a Hamiltonian given as a nonlinear function of the total spin operator S, defined as the sum of the N individual spin operators. The basic idea behind this proposal is the mathematical fact that any non-correlated states can be expanded into eigenstates of S2 and Sz. The same eigenvalues appear several times in the decomposition when N > 2 but all these states differ by their symmetry properties under the N-spin permutations. The case N = 4 and Sz = 1 is treated explicitly and a maximum entanglement is achieved.

Calculation of local spins for correlated wave functions

The recent formula for decomposing the expectation value <S(2)> of the total spin operator for general (correlated) wave functions has been rewritten in terms of the cumulant and is realized numerically for the first time. The results confirm its conformity with the physical expectations.

Mathematical calculation for exact solutions of infinitely strongly interacting Fermi gases in tight waveguides

Exact analytical solutions of the fundamental systems of N quasi-one-dimensional spin-1/2 fermions with infinite delta repulsion in an arbitrary confining potential were presented in our previous letter (Guan et al 2009 Phys. Rev. Lett. 102 160402). The solutions are the simultaneous eigenstates of the Hamiltonian H and the total spin operators S2 and Sz, which fulfil Girardeau's hard-core contacting boundary condition and are antisymmetric under odd permutations among fermions. They are approximate solutions when the coefficient in the delta repulsion is large but finite, and according to the Lieb–Mattis theorem, the solution with the lowest S is the ground state for the system. A detailed mathematical calculation for the exact solutions is presented. The property of the spin-dependent reduced one-body density is discussed.

Local spins: An improved treatment for correlated wave functions

The expectation value 〈 S ^ 2 〉 of the total spin operator for general (correlated) wave functions is transformed to a form in which it appears a physically reasonable sum of atomic and diatomic contributions. The general formula obtained reduces to that given previously in the single determinant case, thus is able to describe in a physically correct manner both limiting cases—the closed shell RHF one which is lacking any spins and the strongly correlated case in which the spins may be considered localized on individual atoms.

Covariant helicity-coupling amplitudes: A new formulation

We have worked out covariant amplitudes for any two-body decay of a resonance with an arbitrary nonzero mass, which involves arbitrary integer spins in the initial and the final states. One key new ingredient for this work is the application of the total intrinsic spin operator which is given directly in terms of the generators of the Poincar\'e group. Using the results of this study, we show how to explore the Lorentz factors which appear naturally, if the momentum-space wave functions are used to form the covariant decay amplitudes. We have devised a method of constructing our covariant decay amplitudes, such that they lead to the Zemach amplitudes when the Lorentz factors are set to one.

Mesonic eightfold way from dynamics and confinement in strongly coupled lattice quantum chromodynamics

We show the existence of all the 36 eightfold way mesons and determine their masses and dispersion curves exactly, from dynamical first principles such as directly from the quark-fluon dynamics. We also give a proof of confinement below the two-meson energy threshold. For this purpose, we consider an imaginary time functional integral representation of a 3+1 dimensional lattice QCD model with Wilson action, SU(3)f global and SU(3)c local symmetries. We work in the strong coupling regime, such that the hopping parameter κ&amp;gt;0 is small and much larger than the plaquette coupling β&amp;gt;1/g02⩾0 (β⪡κ⪡1). In the quantum mechanical physical Hilbert space H, a Feynman-Kac type representation for the two-meson correlation and its spectral representation are used to establish an exact rigorous connection between the complex momentum singularities of the two-meson truncated correlation and the energy-momentum spectrum of the model. The total spin operator J and its z-component Jz are defined by using π∕2 rotations about the spatial coordinate axes, and agree with the infinitesimal generators of the continuum for improper zero-momentum meson states. The mesons admit a labelling in terms of the quantum numbers of total isospin I, the third component I3 of total isospin, the z-component Jz of total spin and quadratic Casimir C2 for SU(3)f. With this labelling, the mesons can be organized into two sets of states, distinguished by the total spin J. These two sets are identified with the SU(3)f nonet of pseudo-scalar mesons (J=0) and the three nonets of vector mesons (J=1,Jz=±1,0). Within each nonet a further decomposition can be made using C2 to obtain the singlet state (C2=0) and the eight members of the octet (C2=3). By casting the problem of determination of the meson masses and dispersion curves into the framework of the the anaytic implicit function theorem, all the masses m(κ,β) are found exactly and are given by convergent expansions in the parameters κ and β. The masses are all of the form m(κ,β=0)≡m(κ)=−2lnκ−3κ2/2+κ4r(κ) with r(0)≠0 and r(κ) real analytic; for β&amp;gt;0,m(κ,β)+2lnκ is jointly analytic in κ and β. The masses of the vector mesons are independent of Jz and are all equal within each octet. All isospin singlet masses are also equal for the vector mesons. For each nonet and β=0, up to and including O(κ4), the masses of the octet and the singlet are found to be equal. But there is a pseudoscalar-vector meson mass splitting given by 2κ4+O(κ6) and the splitting persists for β&amp;gt;0. For β=0, the dispersion curves are all of the form w(p⃗)=−2lnκ−3κ2∕2+(14)κ2∑j=132(1−cospj)+κ4r(κ,p⃗), with ∣r(κ,p⃗)∣⩽const. For the pseudoscalar mesons, r(κ,p⃗) is jointly analytic in κ and pj, for ∣κ∣ and ∣Impj∣ small. We use some machinery from constructive field theory, such as the decoupling of hyperplane method, in order to reveal the gauge-invariant eightfold way meson states and a correlation subtraction method to extend our spectral results to all He, the subspace of H generated by vectors with an even number of Grassmann variables, up to near the two-meson energy threshold of ≈−4lnκ. Combining this result with a previously similar result for the baryon sector of the eightfold way, we show that the only spectrum in all H≡He⊕Ho (Ho being the odd subspace) below ≈−4lnκ is given by the eightfold way mesons and baryons. Hence, we prove confinement up to near this energy threshold.

Eightfold way from dynamical first principles in strongly coupled lattice quantum chromodynamics

We obtain from first principles, i.e., from the quark-gluon dynamics, the Gell’Mann-Ne’eman baryonic eightfold way energy momentum spectrum exactly in an imaginary-time functional integral formulation of strongly coupled lattice quantum chromodynamics in 3+1 dimensions, with local SU(3)c gauge and global SU(3)f flavor symmetries. We take the hopping parameter κ and the pure gauge coupling β satisfying the strong coupling regime condition 0⩽β⪡κ⪡1. The form of the 56 baryon fields emerges naturally from the dynamics and is unveiled using the hyperplane decoupling method. There is no a priori guesswork. In the associated physical quantum mechanical Hilbert space H, spectral representations are derived for the two-baryon functions, which are used to rigorously detect the particles in the energy-momentum spectrum. Using the SU(3)f symmetry, the 56 baryon states admit a decomposition into 8×2 states associated with a spin 1∕2 octet and 10×4 states associated with a spin 3∕2 decuplet. The states are labeled by the quantum numbers of total hypercharge Y, total isospin I, its third component I3, and the value of the quadratic Casimir of SU(3)f; they also carry a label of total spin J and its z component Jz. The total spin operators are defined using π∕2 rotations about the spatial coordinate axes and for improper zero momentum baryon states agree with the infinitesimal generators of the continuum. We show there is a partial restoration of continuous rotational symmetry which implies that all the octet (decuplet) masses are the same. For β=0, the masses of the 56 baryon states have the form M=−3lnκ−3κ3∕4+κ6r(κ), with r(κ) analytic. There is no mass splitting within the octet (decuplet). However, we find an octet-decuplet mass splitting given by 3κ6∕4+O(κ7). For β=0,[M(κ,β)−(−3lnκ)], the non-singular part of the masses, is analytic in κ and β and the mass splitting persists for β≠0. For spatial momentum p≠0, p=(p1,p2,p3)∊(−π,π]3, the 56 baryon dispersion curves have the form w(κ,p)=−3lnκ−3κ3∕4+κ3∑j=1,2,3(1−cospj)∕4+r(κ,p), where r(κ,p) is of O(κ6). For the octet, r(κ,p) is jointly analytic in κ and in each pj for small ∣Impj∣. For each baryon, there is an antibaryon related to it by charge conjugation and with identical spectral properties. It is shown that the spectrum associated with baryons and antibaryons is the only spectrum in the subspace of H with an odd number of quarks, up to near the meson-baryon energy threshold of ≈−5lnκ. A new time reflection is found which is used to define a local spin flip symmetry. The spin flip symmetry, together with the usual parity, time reversal, and spatial π∕2 rotation symmetries and analytic implicit function arguments, are used to obtain these results. Our method extends to the SU(N)f case to uncover (2N+2)!∕[3!(2N−1)!] baryon states and also to treat mesons. Coupling our baryon results with our similar results for the eightfold mesons (of asymptotic mass −2lnκ) shows that the model exhibits confinement up to near the two-meson threshold.

Dynamical eightfold way mesons in strongly coupled lattice QCD

We consider a $3+1$ lattice QCD model with three quark flavors, local $\mathrm{SU}(3{)}_{c}$ gauge symmetry, global $\mathrm{SU}(3{)}_{f}$ flavor symmetry, in an imaginary-time formulation and with strong coupling (a small hopping parameter $\ensuremath{\kappa}&gt;0$ and a plaquette coupling $\ensuremath{\beta}&gt;0$, $0&lt;\ensuremath{\beta}\ensuremath{\ll}\ensuremath{\kappa}\ensuremath{\ll}1$). Associated with the model there is an underlying physical quantum mechanical Hilbert space $\mathcal{H}$ which, via a Feynman-Kac formula, enables us to derive spectral representations for correlations and obtain the low-lying energy-momentum spectrum exactly. Using the decoupling of hyperplane method and concentrating on the subspace ${\mathcal{H}}_{e}\ensuremath{\subset}\mathcal{H}$ of vectors with an even number of quarks, we obtain the one-particle spectrum showing the existence of 36 meson states from dynamical first principles, i.e. directly from the quark-gluon dynamics. The particles are detected by isolated dispersion curves $w(\stackrel{\ensuremath{\rightarrow}}{p})$ in the energy-momentum spectrum. Besides the $\mathrm{SU}(3{)}_{f}$ quantum numbers (total hypercharge, quadratic Casimir ${C}_{2}$, total isospin and its 3rd component), the basic excitations also carry spin labels. The total spin operator $J$ and its $z$-component ${J}_{z}$ are defined using $\ensuremath{\pi}/2$ rotations about the spatial coordinate axes and agree with the infinitesimal generators of the continuum for improper zero-momentum meson states. The eightfold way meson particles are given by linear combinations of these 36 states and can be grouped into three $\mathrm{SU}(3{)}_{f}$ nonets associated with the vector mesons ($J=1$, ${J}_{z}=0$, $\ifmmode\pm\else\textpm\fi{}1$) and one nonet associated with the pseudoscalar mesons ($J=0$). Each nonet admits a further decomposition into a $\mathrm{SU}(3{)}_{f}$ singlet (${C}_{2}=0$) and an octet (${C}_{2}=3$). For $\ensuremath{\beta}=0$, the particle dispersion curves are all of the form $w(\stackrel{\ensuremath{\rightarrow}}{p})=\ensuremath{-}2\mathrm{ln}\ensuremath{\kappa}\ensuremath{-}3{\ensuremath{\kappa}}^{2}/2+(1/4){\ensuremath{\kappa}}^{2}\ensuremath{\sum}_{j=1}^{3}2(1\ensuremath{-}\mathrm{cos}{p}^{j})+{\ensuremath{\kappa}}^{4}r(\ensuremath{\kappa},\stackrel{\ensuremath{\rightarrow}}{p})$, with $\stackrel{\ensuremath{\rightarrow}}{p}\ensuremath{\in}(\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi}{]}^{3}$ and $|r(\ensuremath{\kappa},\stackrel{\ensuremath{\rightarrow}}{p})|\ensuremath{\le}\mathrm{const}$. For the pseudoscalar mesons, $r(\ensuremath{\kappa},\stackrel{\ensuremath{\rightarrow}}{p})$ is jointly analytic in $\ensuremath{\kappa}$ and ${p}^{j}$, for $|\ensuremath{\kappa}|$ and $|\mathcal{I}\mathrm{m}{p}^{j}|$ small. At $\ensuremath{\beta}=0$ the meson masses are given by $m(\ensuremath{\kappa})=\ensuremath{-}2\mathrm{ln}\ensuremath{\kappa}\ensuremath{-}3{\ensuremath{\kappa}}^{2}/2+{\ensuremath{\kappa}}^{4}r(\ensuremath{\kappa})$, with $r(0)\ensuremath{\ne}0$ and $r(\ensuremath{\kappa})$ real analytic; for $\ensuremath{\beta}\ensuremath{\ne}0$ the nonsingular part of the mass, $m(\ensuremath{\kappa},\ensuremath{\beta})+2\mathrm{ln}\ensuremath{\kappa}$, is jointly analytic in $\ensuremath{\kappa}$ and $\ensuremath{\beta}$. For a fixed nonet, the mass of the vector mesons are independent of ${J}_{z}$ and are all equal within each octet. All singlet masses are also equal for the vector mesons. For $\ensuremath{\beta}=0$, up to and including $\mathcal{O}({\ensuremath{\kappa}}^{4})$, for each nonet, the masses of the octet and the singlet are found to be equal. All members of each octet have identical dispersions. Other dispersion curves may differ. Indeed, there is a pseudoscalar, vector meson mass splitting (between $J=0$ and $J=1$) given by $2{\ensuremath{\kappa}}^{4}+\mathcal{O}({\ensuremath{\kappa}}^{6})$ at $\ensuremath{\beta}=0$, analytic in $\ensuremath{\beta}$ and the splitting persists for $\ensuremath{\beta}\ensuremath{\ll}\ensuremath{\kappa}$. Using a correlation subtraction method, we show the 36 meson states give the only spectrum in ${\mathcal{H}}_{e}$ up to near the two-meson threshold of $\ensuremath{\approx}\ensuremath{-}4\mathrm{ln}\ensuremath{\kappa}$. Combining our present result with a similar one for baryons (of asymptotic mass $\ensuremath{-}3\mathrm{ln}\ensuremath{\kappa}$) shows that the model does exhibit confinement up to near the two-meson threshold.

Representation of orthofermion spin operators

The algebraic expressions for the total spin operators expressed in terms of orthofermion creation and annihilation operators are combined into a single equation. The terms in this expressions are rearranged to provide representations of local spin operators. This task is essential for modelling a system of orthofermions in the presence of a magnetic field. By factorizing the orthofermion annihilation (creation) operators into charge and spin dependent parts, it is shown that the latter part suffices to represent spin number, raising and lowering operators. Finally a connection is provided between the spin systems and Greenberg’s infinite statistics.