The irreducible representation under for spin clusters of identical Ii ≥ 1 spin systems are explicitly derived over |IMα> Hilbert space for the range of F z subspaces by using the combinatorial aspects of each ‘weight’ set (M 1 – M 4). This defines the invariance properties for each specific Fz , and consequently for the complete system over ∑Fz . Our investigations show that there are limiting , i.e. independent of Ii , for Fz > Izi for integer spins, or Fz > 2Izi , for half integer spins. Similar considerations apply to the irreducible representations that characterize like ki - | kqv≫ Liouville space for q > (k max – ki ), where k and ki refer to the tensor ranks of the total system and the Ii spin, respectively. The invariance properties of clusters under for I = 1, 3/2 and 5/2 are derived for both | IMα> and | kqv≫ spaces. A general {[Xtilde] i } form over the total Liouville space is given for arbitrary Ii . In the context of the multiquantum N.M.R. of spin clusters, these properties allow one to construct a density operator strictly under the dual groups; such tensors provide a valid -partitioned quantum-Liouville description, of both evolution and relaxation processes, which is applicable to non-magnetically equivalent spin systems, such as [AX]4 under . For bicluster systems specific aspects of recoupling theory, implicit in the evaluation of reduced matrix elements (RME), are examined; explicit basis and RME forms under are considered. Combinatorics are shown to play a large role in characterizing the various spaces, |IMα: Sn >, | kqv: Sn ≫ and , by allowing direct evaluations of the pertinent invariances and components under .
The irreducible representation under for spin clusters of identical Ii ≥ 1 spin systems are explicitly derived over |IMα> Hilbert space for the range of F z subspaces by using the combinatorial aspects of each ‘weight’ set (M 1 – M 4). This defines the invariance properties for each specific Fz , and consequently for the complete system over ∑Fz . Our investigations show that there are limiting , i.e. independent of Ii , for Fz > Izi for integer spins, or Fz > 2Izi , for half integer spins. Similar considerations apply to the irreducible representations that characterize like ki - | kqv≫ Liouville space for q > (k max – ki ), where k and ki refer to the tensor ranks of the total system and the Ii spin, respectively. The invariance properties of clusters under for I = 1, 3/2 and 5/2 are derived for both | IMα> and | kqv≫ spaces. A general {[Xtilde] i } form over the total Liouville space is given for arbitrary Ii . In the context of the multiquantum N.M.R. of spin clusters, these properties allow one to construct a density operator strictly under the dual groups; such tensors provide a valid -partitioned quantum-Liouville description, of both evolution and relaxation processes, which is applicable to non-magnetically equivalent spin systems, such as [AX]4 under . For bicluster systems specific aspects of recoupling theory, implicit in the evaluation of reduced matrix elements (RME), are examined; explicit basis and RME forms under are considered. Combinatorics are shown to play a large role in characterizing the various spaces, |IMα: Sn >, | kqv: Sn ≫ and , by allowing direct evaluations of the pertinent invariances and components under .
For many-body spin cluster problems, dual-symmetry recoupled tensors over Liouville space provide suitable bases for a generalized torque formalism using the 𝒮n-adapted density operator in which to discuss NMR and related techniques. The explicit structure of such tensors is considered in the context of the Cayley algebra of scalar invariants over a field, specified by the inner ki rank labels of the Tkq(kl−kn)s. The pertinence of both lexical combinatorial architectures over inner rank sets and SU2 propagative topologies in specifying the structure of dual recoupling tensors is considered in the context of the 𝒮n partitional aspects of spin clusters. The form of Heisenberg superoperator generators whose algebra underlies the Gel’fand pattern algebra of SU(2) and SU(2)×𝒮n tensor bases over Liouville space is presented together with both the related 𝓈-boson algebras and a description of the associated {‖2k 0〉〉} pattern sets of CF29H carrier space under the appropriate symmetry. These concepts are correlated with recent work on SU(2)×𝒮n induced symmetry hierarchies over Liouville spin space. The pertinence of this theoretical work to an understanding of multiquantum NMR in Liouville space formalisms is stressed in a discussion of the nature of pathways for intracluster J coupling, which also gives a valuable physical insight into the nature of coherence transfer in more general spin-1/2 systems.
Analytic SU(2) × Sn dual tensorial (DT) spin dynamics over uniform NMR spins is invoked in examining the modern quantum basis for the universal non-observability rule which governs dominant intracluster JXX' couplings of (A)[X]n NMR systems as a specific form of (abstract) permutational spin symmetry (PSS) with well defined properties on spin-alone space. This is shown to be linked to DT constraints that apply to the cross-product ϕ1±1(1.1) polarisation development i.e., as being confined to [1n](Sn) (Liouvillian) salient, with the existence of [n] (rotating frame) null subspaces. Both these arise within the spin dynamics of (A)[X]2 spin systems (or subsystems thereof) within (a hierarchy of) dominant JXX' governing the internal L(0); such spin systems provide analytic sequels to comparative spin dynamics studies of XX' PSS and AX broken-PSS systems in a Liouvillian coupled tensorial basis formalism, since both draw on (Sanctuary B. C.: Mol. Phys. 1985, 55, 1017), and on the realisation that proper PSS over a (uniform) spin-space L(0) = [H(0),.] zeroth-order Liouvillian and its internal (hierarchical subsets of) JXX' (Temme F. P.: J. Mol. Struct. (THEOCHEM) 2002, 547, 153) i.e., for abstract Sn ↓ G group embeddings. The present work also examines the general irrep-structure of DT spin symmetries for the extent of unit-character irreps and the role of Sn ⊃ Sn-1 ⊃ .. ⊃ [2](S2) group chains in defining the Sn multiple invariants under democratic recoupling of PSS of uniform spin systems. As group measures, these properties apply to both (A)[X]n and [AX]n PSS symmetries, with the invariant cardinality |SI|(n) being related to time-reversal invariance (TRI) and its inherent democratic recoupling (DR) over Weyl (I • I) pairs. For [X]2n uniform spin clusters, |SI|(2n) is best derived via n-fold polyhedral combinatorics of the underlying DR (Temme F. P.: Proc. R. Soc. London, Ser. A 2005, 461, 321) i.e., as an augmented post-Weyl view of the essential role of TRI in (group) invariant cardinality, with the Sn-invariants represented by certain Sn subduction properties.
Augmented quasiparticle (QP) mappings, as applied to indistinguishable point sets of (Liouvillian) democratic-recoupled (DR) tensors, provide for a 1:1 invariant labelling of the underlying (disjoint) dual projective map carrier subspaces, where the Liouville pattern basis set is defined via superboson unit-tensor actions on a null space, |∅⟩⟩. The co-operative-action Liouville algebras described here imply parallel limitations to Jucys graph recoupling and its related Racah–Wigner (R–W) algebras once DR indistinguishable point tensorial sets are involved, as in non-SR dominant (NMR) spin symmetry. The importance of G-invariants, as labels for disjoint carrier subspaces in such automorphic spin symmetries, arises from their essential role in defining the quantal-completeness of indistinguishable point sets. From the established properties of augmented-QPs as super-bosons (Temme 2002 Int. J. Quantum Chem. 89 429) (i.e., beyond the earlier Hilbert-space-based Louck and Biedenharn boson pattern views), insight into Atiyah and Sutcliffe's (A–S) assertions (Atiyah and Sutcliffe 2002 Proc. R. Soc. A 458 1089) on the limitations of graph recoupling theory to distinct point sets is obtained. This clarifies the wider analytic intractible of automorphic DR spin systems—beyond the Lévi–Civitá cyclic-commutation (R–W) approach (Lévy-Leblond and Lévy-Nahas 1965 J. Math. Phys. 6 1372) which holds for a mono-invariant problem. For (rotating-frame) density matrix approaches to [ A ] n , [ A ] n ( X ) and (dual) NMR systems, the focus is necessarily on the specialized nature of indistinguishable point sets within multiple invariant-theoretic-based, dynamical spin physics. Here, the GI(s) (GI-cardinality) constitute an important part of the dual irrep set, , with combinatorics, as a central facet of invariant theory, playing a crucial role in the concept of 'quantal completeness' and the impact of A–S's assertion on the existing NMR tensorial physics. Clearly, the role of Liouvillian Yamanouchi projection, now as disjoint subspatial-based transformational properties, defines such DR bases and their unit tensors. A brief outline is given of the structure of augmented general-indexed Lévi–Civitá superoperators with their multiple GI-labelled carrier subspaces.
We present two methods to study model Hamiltonians defined on clusters. One is the symmetrization of the Hilbert space basis according to irreducible representations of the symmetry group of the Hamiltonian. The other one adapts the Lanczos diagonalization method directly to the second quantized form of the Hamiltonian studied. By the symmetrization of the basis, one takes advantage of the symmetries of the Hamiltonian, and uses the group theory to find the invariant subspaces of the Hamiltonian in the complete Hilbert space. Then one needs only to work within the invariant subspaces. This method cuts down substantially the size of the matrix representation of the Hamiltonian to be diagonalized. Therefore, one could study relatively larger systems. As a result, all the eigenvectors and eigenvalues are obtained. Consequently, one could calculate thermodynamical quantities and various correlation functions, such as, the spin correlation, the charge correlation, the superconductivity pairing correlation, and the photoemission spectra. One the other hand, the Lanczos method, mentioned here in particular, is a specially developed computer program for the study of the model Hamiltonians. This program uses the key feature of the Lanczos method in diagonalization, allows one to use the second quantized form of the Hamiltonian directly without pregeneration or storing the matrix elements of the Hamiltonian. Two methods could be combined to take the full advantage of the symmetries of the Hamiltonian by input of the symmetrized wavefunction as the starting trial wavefunction in the Lanczos method. Here we illustrate the above two methods by applying to the Hubbard, and extended Hubbard models on a simple cubic cluster, and the lattice Anderson model on the two dimensional square lattice. Other interesting results for different systems are also briefly discussed here.
For S n –based, uniform auxiliary dual tensorial (UDT) sets, explicitly with even maximal all–alike j i ( k i ) over v ≡ ( j 1 ⋯ j 2 n ), or ( k 1 ⋯ k 2 n ), sub–rank auxiliary labelling, the specific time–reversal invariance–dependent dual group invariants and their independent cardinalities (| SI | (2 n ) ) are fundamental to understanding simply reducible (SR) dynamical spin system structure over (Liouvillian) carrier space, a quantum physics property implicit in dual quasi–particle (QP) formalisms associated with, for example, uniform identical multiple spin NMR systems . Such UDT sets stand in total contrast to the distinct j i ( k i ) of ( j 1 ⋯ j″ n )(( k 1 ⋯ k″ n )) sub–rank orthogonal tensorial sets with their Z aa′ graph invariants which for reasons discussed in the text are of restricted validity. For UDTs as operator bases related to the (spin) interaction network–based tensors of high S n permutational NMR Hamiltonians (Liouvillians), the use of QP boson (superboson) formalisms and their dual projective maps (F. P. Temme 1993 Physica A 198 , 245–261) prove invaluable. Once the independent cardinality of the multiple invariants of these UDTs (alias the SU (2) × S 2 n group invariants) (here denoted | SI | (2 n ) ) are known in terms of democratic re–coupling (DR) and polyhedral combinatorial (PC) sampling applied to the time–reversal invariance (TRI) properties of the uniform multiple spin (sub–)system, so the SR dual carrier spaces , ℍ˜, ℍ˜ v follow directly as well–defined entities. For UDTs of general (2 n > 6)–fold ensembles, the augmented models of TRI discussed here (i.e. beyond linear re–coupled Weyl forms) are essential, due to the presence of non–Abelian degeneracy. A regular geometric automorphic limit is established for sequential (2 n )–based | SI | (2 n ) total series, specific to (higher) uniform ( j 1 ⋯ j i ), ( k 1 ⋯ k i )–based dual tensors. By treating the | SI | (2 n ) obtained from the DR–based TRIs as group measures and invoking an automorphic geodesic augmentation, certain otherwise inaccessible, additional | SI | (2 n′ ) (and hence UDTs) may be established; these are based on sporadic higher (regular) (2 n ≫ 12)–fold ensemble invariants and are obtained via finite group duality properties implicit in the concept of group measures. The PC DR–based views of general (2 n ) interacting spins of UDTs expressed here give specific insight into dynamic structure over (explicit TRI invariant–defined) Liouvillian carrier spaces. This lattice point (Erdösian) view of DR shows that UDT invariants are distinct and more contracted than predicted by the earlier linear–re–coupling–based | D 0 ( U )|(⨷ SU (2)) approach. The presence of degeneracy in general high (2 n ) UDTs as non–Abelian entities ensures that the homomorphic–based, | D 0 ( U )| enumeration (with its implied linear graphical re–coupling) is invalid for high (2 n ) uniform (NMR) multi–spin systems whose TRI properties are governed by DR re–coupling. The prime focus of the report concerns the positive conceptual value of PC lattice–point–based DR, as applied to the S 2 n –invariants which define UDTs (of NMR) and their completeness. Brief contrasts with the earlier, more restricted O (2 k + 1) (overall) invariants of ( O ( n ) ⊃ ⋯ ⊃ G )–based democratic re–coupled quantum systems are given for completeness.
Structural aspects of superboson mappings and their dual group-based carrier spaces inherent in quantum-Liouville NMR formalisms are presented for their conceptual role in understanding the transformational properties and the spin dynamics of identical [A]n([Formula: see text]n) spin ensembles or [AX]n systems, given in terms of {Tkq (υ)} º {|kqυ > >} tensorial bases. Interest in the explicit democratic labelling of the Liouvillian carrier subspaces of [A]n spin ensembles, prompts an examination of the inner products (ITPs) that define the projective carrier space associated with superbosons. A weak-branching (WB) limit (for (bipartite) partitions λ [Formula: see text] n, for n-indexed (SU2 ×)[Formula: see text]n) ITPs gives rise to: (i) maximal coefficient sets for [µ] [Formula: see text] [µ']([Formula: see text]n) irrep products (in Butler's notation), under a sufficiently high (n [Formula: see text] 4µ, 4µ') indexed [Formula: see text]n group, or (ii) to identical numerical {c[Formula: see text], λ' } sequence-ordered sets of reduction coefficients over similar, displaced-sequence fields on augmenting ITP to [µ] [Formula: see text] [µ''], where µ'' > µ' > µ, or else (iii) to some (sequence-displaced) subset of the latter. The origins of the decompositional WB limit as part of group structure may be traced to an algorithmic similarity between the LittlewoodRichardson and Young(III) combinatorial rules. Alternative approaches to the bipartite product decompositional mappings are possible, using both (subspatial-restricted) Schur-function techniques of Wybourne and [Formula: see text]16[Formula: see text]n[Formula: see text]28 symbolic algorithmic enumerations based on the SYMMETRICA discrete-maths. package of Kerber et al. (J. Symbolic Comput. 14, 195 (1993)). The nature of maximal sets and scaling in ITP decompositions is established and recognition given to the role of combinatorics, [Formula: see text]n algorithms and the superboson algebras of the SU(2) × [Formula: see text]n group in (multiple) quantized spin physics. PACS Nos.: 02.10, 03.65Ca, 33.20Vq, 33.25+k