Spin Projectors Research Articles

Projective Spin Adaptation for the Exact Diagonalization of Isotropic Spin Clusters

Spin Hamiltonians, like the Heisenberg model, are used to describe the magnetic properties of exchange-coupled molecules and solids. For finite clusters, physical quantities, such as heat capacities, magnetic susceptibilities or neutron-scattering spectra, can be calculated based on energies and eigenstates obtained by exact diagonalization (ED). Utilizing spin-rotational symmetry SU(2) to factor the Hamiltonian with respect to total spin S facilitates ED, but the conventional approach to spin-adapting the basis is more intricate than selecting states with a given magnetic quantum number M (the spin z-component), as it relies on irreducible tensor-operator techniques and spin-coupling coefficients. Here, we present a simpler technique based on applying a spin projector to uncoupled basis states. As an alternative to Löwdin’s projection operator, we consider a group-theoretical formulation of the projector, which can be evaluated either exactly or approximately using an integration grid. An important aspect is the choice of uncoupled basis states. We present an extension of Löwdin’s theorem for s=12 to arbitrary local spin quantum numbers s, which allows for the direct selection of configurations that span a complete, linearly independent basis in an S sector upon the spin projection. We illustrate the procedure with a few examples.

Analytical Solutions of Symmetric Isotropic Spin Clusters Using Spin and Point Group Projectors

Spin models like the Heisenberg Hamiltonian effectively describe the interactions of open-shell transition-metal ions on a lattice and can account for various properties of magnetic solids and molecules. Numerical methods are usually required to find exact or approximate eigenstates, but for small clusters with spatial symmetry, analytical solutions exist, and a few Heisenberg systems have been solved in closed form. This paper presents a simple, generally applicable approach to analytically solve isotropic spin clusters, based on adapting the basis to both total spin and point group symmetry to factor the Hamiltonian matrix into sufficiently small blocks. We demonstrate applications to small rings and polyhedra, some of which are straightforward to solve by successive spin-coupling for Heisenberg terms only; additional interactions, such as biquadratic exchange or multi-center terms necessitate symmetry adaptation.

Softly higher-derivative massive gravity

In this work, we study the higher-derivative massive gravity in D-dimensional space-time with the mass term arisen due to a 2-brane embedded in a 4D Minkowski spacetime. We consider the effect of a resonance mass term from the DGP braneworld model for the higher-derivative massive gravity. Moreover, we add the gravitational Chern–Simons and Ricci–Cotton terms to this model and evaluate the graviton propagator using a basis of Barnes–Rivers spin projectors. Using the obtained propagator, we proceed with discussing the consistency of the model, writing the dispersion relations, and analyzing causality and unitarity.

Metric-Affine Gravity as an effective field theory

We discuss theories of gravity with independent metric (or frame field) and connection, from the point of view of effective field theory. We count the parity-even Lagrangian terms of dimension up to four and give explicit bases for the independent terms that contribute to the two-point function. We then give the decomposition of the linearized action on a complete basis of spin projectors and consider various subclasses of Metric-Affine Gravity (MAG) theories. We show that teleparallel theories can be dynamically equivalent to any metric theory of gravity and give the particle content of those whose Lagrangian contains only dimension-two terms. We point out the existence of a class of MAGs whose equations of motion do not admit propagating degrees of freedoms. Finally, we construct simple MAGs that contain only a massless graviton and a state of spin/parity 2− or 3−. As a side result, we write the relativistic wave equation for a spin/parity 2− state.

Unified structure for exact towers of scar states in the Affleck-Kennedy-Lieb-Tasaki and other models

Quantum many-body scar states are many-body states with finite energy density in non-integrable models that do not obey the eigenstate thermalization hypothesis. Recent works have revealed "towers" of scar states that are exactly known and are equally spaced in energy, specifically in the AKLT model, the spin-1 XY model, and a spin-1/2 model that conserves number of domain walls. We provide a common framework to understand and prove known exact towers of scars in these systems, by evaluating the commutator of the Hamiltonian and a ladder operator. In particular we provide a simple proof of the scar towers in the integer-spin 1d AKLT models by studying two-site spin projectors. Through this picture we deduce a family of Hamiltonians that share the scar tower with the AKLT model, and also find common parent Hamiltonians for the AKLT and XY model scars. We also introduce new towers of exact states, organized in a "pyramid" structure, in the spin-1/2 model through successive application of a non-local ladder operator.

Parity violation and modification of fermion spin projectors

Account of interactions in frameworks of Standard Model leads to appearence of γ5 terms in a dressed fermion propagator. In this case the standard spin projectors do not commute with propagator and should be modified. Starting point of our discussion is the eigenvalue problem for inverse propagator S(p). Having the solutions of this problem, one can understand the necessary modification of spin projectors. The same generalized spin projector appears at propagation of neutrino in a moving matter, but in contrast to vacuum case, corresponding polarization vector is not arbitrary. This spin projector commute with neutrino propagator and all dispersion laws for neutrino in media are classified according to spin projection onto this fixed axis.

Spin projection operators in (A)dS and partial masslessness

We elaborate on the traceless and transverse spin projectors in four-dimensional de Sitter and anti-de Sitter spaces. The poles of these projectors are shown to correspond to partially massless fields. We also obtain a factorisation of the conformal operators associated with gauge fields of arbitrary Lorentz type (m/2,n/2), with m and n positive integers.

Neutrino propagation in media and axis of complete polarization

We construct a spectral representation of neutrino propagator in moving matter or in external magnetic field. In both cases there exists fixed four-dimensional axis of polarization, such that the corresponding spin projectors commute with propagator. As a result, all eigenvalues of propagator and, consequently, dispersion laws for neutrino in media are classified according to spin projection onto this axis. Use of the found spin projectors simplifies essentially the eigenvalue problem and allows to build spectral representation of propagator in moving matter or external magnetic field in analogy with the vacuum propagator.

Off-shell fermion polarization and t-quark production

Standard calculation of the polarization of final electron for pure initial state may be reformulated as a problem of looking for the complete polarization axis of produced state. It gives method for calculation of polarization applicable for both final and intermediate state fermions. We discuss modification of the energy and spin projectors in theory with parity violation. The obtained projectors are used to give the most accurate parametrization of t-quark resonance curve and simultaneously for its off-shell polarization.

Neutrino propagator in media: spin properties and spectral representation

We have found that in case of propagation of neutrino in the moving matter or external magnetic field there is a fixed 4-axis of polarization, which corresponds to spin projectors commute with a propagator. It means that, states with the definite spin projection on this axis in media have a definite dispersion law. The presence of this axis essentially simplifies the eigenvalue problem and allows one to build spectral representation of the propagator in media.

Understanding close-lying exotic charmonia states within QCD sum rules

Motivated by the experimental findings of some new exotic states decaying into channels like J/ψϕ, we investigate the formation of resonances/bound states in the Ds⁎D¯s⁎ system using QCD sum rules. To do this we start with a current of the type vector times vector and use spin projectors to separate the spin 0, 1 and 2 contributions to the correlation function. We find three states with isospin 0, nearly spin degenerate, with a mass around 4.1 GeV. We have also investigated the decay of these states to J/ψϕ and provide the corresponding partial widths. Such information should be useful for experimental studies in future.

Spin-Projected Matrix Product States: Versatile Tool for Strongly Correlated Systems.

We present a new wave function ansatz that combines the strengths of spin projection with the language of matrix product states (MPS) and matrix product operators (MPO) as used in the density matrix renormalization group (DMRG). Specifically, spin-projected matrix product states (SP-MPS) are constructed as [Formula: see text], where [Formula: see text] is the spin projector for total spin S and |ΨMPS(N,M)⟩ is an MPS wave function with a given particle number N and spin projection M. This new ansatz possesses several attractive features: (1) It provides a much simpler route to achieve spin adaptation (i.e., to create eigenfunctions of Ŝ2) compared to explicitly incorporating the non-Abelian SU(2) symmetry into the MPS. In particular, since the underlying state |ΨMPS(N,M)⟩ in the SP-MPS uses only Abelian symmetries, one does not need the singlet embedding scheme for nonsinglet states, as normally employed in spin-adapted DMRG, to achieve a single consistent variationally optimized state. (2) Due to the use of |ΨMPS(N,M)⟩ as its underlying state, the SP-MPS can be closely connected to broken-symmetry mean-field states. This allows one to straightforwardly generate the large number of broken-symmetry guesses needed to explore complex electronic landscapes in magnetic systems. Further, this connection can be exploited in the future development of quantum embedding theories for open-shell systems. (3) The sum of MPOs representation for the Hamiltonian and spin projector [Formula: see text] naturally leads to an embarrassingly parallel algorithm for computing expectation values and optimizing SP-MPS. (4) Optimizing SP-MPS belongs to the variation-after-projection (VAP) class of spin-projected theories. Unlike usual spin-projected theories based on determinants, the SP-MPS ansatz can be made essentially exact simply by increasing the bond dimensions in |ΨMPS(N,M)⟩. Computing excited states is also simple by imposing orthogonality constraints, which are simple to implement with MPS. To illustrate the versatility of SP-MPS, we formulate algorithms for the optimization of ground and excited states, develop perturbation theory based on SP-MPS, and describe how to evaluate spin-independent and spin-dependent properties such as the reduced density matrices. We demonstrate the numerical performance of SP-MPS with applications to several models typical of strong correlation, including the Hubbard model, and [2Fe-2S] and [4Fe-4S] model complexes.

Mixing of fermions and spectral representation of propagator

We develop the spectral representation of propagator for [Formula: see text] mixing fermion fields in the case of [Formula: see text]-parity violation. The approach based on the eigenvalue problem for inverse matrix propagator makes possible to build the system of orthogonal projectors and to represent the matrix propagator as a sum of poles with positive and negative energies. The procedure of multiplicative renormalization in terms of spectral representation is investigated and the renormalization matrices are obtained in a closed form without the use of perturbation theory. Since in theory with [Formula: see text]-parity violation the standard spin projectors do not commute with the dressed propagator, they should be modified. The developed approach allows us to build the modified (dressed) spin projectors for a single fermion and for a system of fermions.

RelatingD*D¯*currents withJπ=0+,1+and2+toZcstates

In this work we study the $D^* \bar{D}^*$ current with QCD sum rules. We write the correlation function using the general current corresponding to the $D^* \bar{D}^*$ system and then use spin projectors to obtain the correlation function in the $0^+, 1^+$ and $2^+$ spin-parity configurations. The purpose of the present work is to investigate the possibility of explaining the recently reported $Z_c(4025)$ as a $D^* \bar{D}^*$ moleculelike state. As a result we find a state for each spin case with a very similar mass: $M^{S=0} = \left(3943 \pm 104\right)$ MeV, $M^{S=1} = \left(3950 \pm 105\right)$ MeV, $M^{S=2} = \left(3946 \pm 104 \right)$ MeV. We discuss that our mass results, within error bars, for $1^+$ or $2^+$ are both compatible with $Z_c(4025)$. However, our results are also compatible with a possible $D^* \bar{D}^*$ bound state, in agreement with predictions of some previous works. We have also calculated the current-state coupling which turns out to be larger in the $2^+$ case.

Exploring theD*ρsystem within QCD sum rules

We present a study of the ${D}^{*}\ensuremath{\rho}$ system made by using the method of QCD sum rules to determine the mass of possible resonances generated in the same system. Using isospin and spin projectors, we investigate the different configurations and obtain evidences for three ${D}^{*}$ mesons with isospin $I=1/2$, spin $S=0$, 1, 2 and with masses $2500\ifmmode\pm\else\textpm\fi{}67$, $2523\ifmmode\pm\else\textpm\fi{}60$, and $2439\ifmmode\pm\else\textpm\fi{}119\text{ }\text{ }\mathrm{MeV}$, respectively. The last state can be associated with ${D}_{2}^{*}(2460)$ (spin 2) listed by the Particle Data Group, while one of the first two might be related to ${D}^{*}(2640)$, with unknown spin parity. In the case of $I=3/2$ we also find evidences of three states with spin 0, 1, and 2, respectively, with masses $2467\ifmmode\pm\else\textpm\fi{}82$, $2420\ifmmode\pm\else\textpm\fi{}128$, and $2550\ifmmode\pm\else\textpm\fi{}56\text{ }\text{ }\mathrm{MeV}$. The results for the sector $I=1/2$ and $S=0$, 1, 2, are intriguingly similar to a previous study of the ${D}^{*}\ensuremath{\rho}$ system based on effective field theories, supporting in this way a molecular picture for the resonances ${D}^{*}(2640)$ and ${D}_{2}^{*}(2460)$, while the results for $I=3/2$ hint towards the existence of exotic mesons since a multiquark configuration is required to get the quantum numbers of the states found.

Automatic generation of quarkonium amplitudes in NRQCD

We present a simple method to automatically evaluate arbitrary tree-level amplitudes involving the production or decay of a heavy quark pair QQbar in a generic {2S+1}L_J^[1,8] state, i.e., the short distance coefficients appearing in the NRQCD factorization formalism. Our approach is based on extracting the relevant contributions from the open heavy quark-antiquark amplitudes through an expansion with respect to the quark-antiquark relative momentum and the application of suitable color and spin projectors. To illustrate the capabilities of the method and its implementation in MadGraph a few applications to quarkonium collider phenomenology are presented. Comment: 17 pages, 7 figures

Geminal operators and symmetric requirements

This paper continues our search for the unification of the spin-free and the spin-dependent theories and their required conditions. First, besides the previously uncovered powerful ability of the spin double symmetrizer in the spin part of the primitive Hartree product, we have extended our examination to the additional role played by the spatial geminal (valence bond) operator in the spatial part of the Hartree product . This has thus completed our studies to reveal the important hidden role of the Hartree product. Secondly, we are able to show the capability of the spatial geminal symmetric requirement (GSR) which is the spin-free counterpart of the well-known spin-dependent spin geminal antisymmetric requirement (GAR), and to prove that these two symmetric requirements are interchangeable and either one of them is sufficient. The conditions for these requirements to be satisfied are given. We have also proved that the operation of spin geminal operator on the primitive spin function product is not necessary as it has no effect under the orthogonal spin projection. Its arising difference and similarity with the two symmetric requirements is shown. The spin GAR satisfactory problem of the Yamanouchi Kotani spin eigenfunctions is also explained. An important message is obtained: not all the spatial properties can be replaced by their equivalent supplementary dual spin functions, and vice versa, although these spatial properties may be passed over through the use of the spatial–spin interchangeability theorem (SSIT). An immediate usage of this SSIT to the primitive Hartree product is shown. Finally, various forms of the spin-free and spin-dependent orthogonal spin projectors are produced including those with λ orbital pairs or λ doubly occupied orbitals; they are also interchangeable and exhibit the one-to-one correspondence between the spin-free and the spin-dependent formalisms through our equivalent transformation rule just as expected. Useful relations between these spin projectors and the antisymmetrizers which are helpful for the evaluation of Hamiltonian matrix elements are given.

Double symmetrization and the linearly independent spin eigenfunctions

AbstractIt is shown that the exclusion principle merely requires the electrons to be divided into two distinct groups, then, according to our assumption of the distinguishability of electrons, its mathematical representation can be the spatial double antisymmetrization or the spatial‐spin antisymmetrization; in the latter case an auxiliary spin double symmetrization is also required in order to distinguish the two types of electrons. The second requirement is used to recognize the fact that, although all electrons can be assumed to be indistinguishable under certain condition, their natural characteristic of two distinct groups must still be preserved.It is further shown that the double antisymmetrization (or double symmetrization) determines the spin quantum number M, whereas the Löwdin spin projector for nonorthogonal eigenfunctions and the Wigner matric basis for orthogonal eigenfunctions will decide the spin quantum number S, Both the spatial double antisymmetrizer and the spin double symmetrizer will reject all the unnecessary eigenfunctions and project out only one unique set of linearly independent eigenfunctions.

Exclusion principle and the symmetry adapted wave‐functions

AbstractWave‐functions of various spin‐dependent and spin‐free methods are examined from the point of view of the requirements of the exclusion principle and the spin projection. It is shown that the “two‐rowed” or “two‐columned” requirements of the standard Young tableaux are necessary but not sufficient to replace the “antisymmetry” requirement of a wave‐function and to be regarded as exclusion principle. The symmetry adapted wave‐functions which are constructed from the matric basis e may not satisfy the exclusion principle and, hence, their usages are open to question. The appropriate symmetry adapted wave‐functions which satisfied the exclusion principle are given for any pure spin state. We have also shown that the structure operators NP for bond functions are spin projectors in the Waller‐Hartree double‐antisymmetrized space only, and should not be used in the Hartree product space. Furthermore, if the corresponding matric operators PNP are used in the Hartree product space, then the wave‐functions thus constructed may not be antisymmetric with respect to the permutations of indistinguishable electrons.

Young operators and the Waller–Hartree spinfree method

AbstractThe basic equivalent transformation Pσ = (−1)p (Pr)−1 as developed recently [6] is used to transform between the Young operators for a two‐row standard tableau and its conjugate two‐column standard tableau. These Young operators are shown to be the Löwdin spin projectors in the spin function‐dependent methods and the spin projectors in the spin function‐free methods. The sandwich operator of Gallup and the last diagonal element of the matric basis of Matsen are shown to be the special cases of our more general matric projectors. Several different forms of the Löwdin spin projectors, spin projectors, and matric projectors are shown and their relations are discussed.