A numerical algorithm for the explicit calculation of SU(<i>N</i>) and $\mbox{SL}(N,\mathbb {C})$SL(N,C) Clebsch–Gordan coefficients

Alex, Arne; Kalus, Matthias; Huckleberry, Alan; Delft, Jan von

doi:10.1063/1.3521562

Cited by 89 publications

(94 citation statements)

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“…This matrix element is related to the channel α L = α R + 0 and therefore to the two-fold degenerate 0 weight in the adjoint representation. The same phenomenon happens in the classical sl 3 representation theory where the SU(3) Clebsch-Gordan coefficients are not uniquely defined when multiplicities are present (see for instance [20]). …”

Section: Jhep06(2016)137mentioning

confidence: 87%

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Correlation functions with fusion-channel multiplicity in W 3 $$ {\mathcal{W}}_3 $$ Toda field theory

Belavin

Estienne

Foda

et al. 2016

J. High Energ. Phys.

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Current studies of W N Toda field theory focus on correlation functions such that the W N highest-weight representations in the fusion channels are multiplicity-free. In this work, we study W 3 Toda 4-point functions with multiplicity in the fusion channel. The conformal blocks of these 4-point functions involve matrix elements of a fullydegenerate primary field with a highest-weight in the adjoint representation of sl 3 , and a fully-degenerate primary field with a highest-weight in the fundamental representation of sl 3 . We show that, when the fusion rules do not involve multiplicities, the matrix elements of the fully-degenerate adjoint field, between two arbitrary descendant states, can be computed explicitly, on equal footing with the matrix elements of the semi-degenerate fundamental field. Using null-state conditions, we obtain a fourth-order Fuchsian differential equation for the conformal blocks. Using Okubo theory, we show that, due to the presence of multiplicities, this differential equation belongs to a class of Fuchsian equations that is different from those that have appeared so far in W N theories. We solve this equation, compute its monodromy group, and construct the monodromy-invariant correlation functions. This computation shows in detail how the ambiguities that are caused by the presence of multiplicities are fixed by requiring monodromy-invariance.

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Section: Jhep06(2016)137mentioning

confidence: 87%

“…From the first relation in (5.19) and from (5.9), the matrix X has to have the following form 20) Up to a global normalization, we have five unknown constants x i /x 1 , i = 2, · · · , 6 to determine. This will be fixed by imposing the second constraint in (5.19).…”

Section: Monodromy-invariant Correlation Functionmentioning

confidence: 99%

Correlation functions with fusion-channel multiplicity in W 3 $$ {\mathcal{W}}_3 $$ Toda field theory

Belavin

Estienne

Foda

et al. 2016

J. High Energ. Phys.

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“…12 to symmetries with outer multiplicities. As an application, we performed a detailed DM-NRG study of the SU(3) symmetrical Anderson model by first incorporating SU(N ) symmetries 16 in the Open Access Budapest DM-NRG code, 13 and then performing the numerical calculations using the complete U(1) × SU(3) symmetry of the model. A similar extension has been carried out within the matrix product state (MPS) approach parallel to this work.…”

Section: Discussionmentioning

confidence: 99%

“…Whereas these are known in closed form for SU (2), this is not the case for SU(N > 2). However, an efficient numerical algorithm for their evaluation has recently been developed, 16 which we use here. This paper is structured as follows: In Sec.…”

Section: Introductionmentioning

confidence: 99%

SU(3) Anderson impurity model: A numerical renormalization group approach exploiting non-Abelian symmetries

et al. 2012

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We show how the density-matrix numerical renormalization group method can be used in combination with nonAbelian symmetries such as SU(N ). The decomposition of the direct product of two irreducible representations requires the use of a so-called outer multiplicity label. We apply this scheme to the SU(3) symmetrical Anderson model, for which we analyze the finite size spectrum, determine local fermionic, spin, superconducting, and trion spectral functions, and also compute the temperature dependence of the conductance. Our calculations reveal a rich Fermi liquid structure.

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“…For N > 2, closed forms for the required coefficients are not known to the authors, but can be calculated efficiently using standard algorithms as in Ref. [54]. In Fig.…”

Section: Smentioning

confidence: 99%

Realizing exactly solvableSU(N)magnets with thermal atoms

et al. 2016

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We show that n thermal fermionic alkaline-earth-metal atoms in a flat-bottom trap allow one to robustly implement a spin model displaying two symmetries: the S n symmetry that permutes atoms occupying different vibrational levels of the trap and the SU(N ) symmetry associated with N nuclear spin states. The symmetries make the model exactly solvable, which, in turn, enables the analytic study of dynamical processes such as spin diffusion in this SU(N ) system. We also show how to use this system to generate entangled states that allow for Heisenberg-limited metrology. This highly symmetric spin model should be experimentally realizable even when the vibrational levels are occupied according to a high-temperature thermal or an arbitrary nonthermal distribution.DOI: 10.1103/PhysRevA.93.051601The study of quantum spin models with ultracold atoms [1,2] promises to give crucial insights into a range of equilibrium and nonequilibrium many-body phenomena from quantum spin liquids [3] and many-body localization [4] to quantum quenches [5][6][7] and quantum annealing [8]. While other approaches exist [9][10][11][12], the most common approach taken to implement a quantum spin model with ultracold atoms relies on preparing a Mott insulator in an optical lattice, where the internal states of atoms on each site define the effective spin [1,[13][14][15][16][17][18][19]. Virtual hopping processes to neighboring sites and back then give rise to effective superexchange spin-spin interactions. Since the superexchange interactions are typically very weak ( kHz) [1] (unless the traps are operated near surfaces, which can reduce spacings and increase energy scales [20][21][22]), it is a significant challenge in experimental cold-atom physics to achieve temperatures and decoherence rates low enough to access superexchange-based quantum magnetism.Since ultracold atoms can be prepared in specific internal (i.e., spin) states with extremely high precision, spin temperatures that can be realized are much lower than the experimentally achievable motional temperatures. It is therefore tempting to circumvent the problem of high motional temperature by constructing a spin model in such a way that the motional and spin degrees of freedom are effectively decoupled. We provide a recipe for such a decoupling and hence for realizing spin models with thermal atoms.The first crucial ingredient for implementing such a spin model is to depart from second-order superexchange interactions and use contact interactions to first order [23][24][25][26][27][28][29][30][31][32]. As shown in Fig. 1(a), this can be achieved if all atoms sit in different orbitals of the same anharmonic trap and remain in these orbitals throughout the evolution, which is a good approximation for weak interactions [23][24][25]30,31]. In that case, the occupied orbitals play the role of the sites of the spin Hamiltonian. However, because of high motional temperature in such systems, every run of the experiment typically yields a different set of populated orbitals and hence a diff...

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A numerical algorithm for the explicit calculation of SU(N) and $\mbox{SL}(N,\mathbb {C})$SL(N,C) Clebsch–Gordan coefficients

Cited by 89 publications

References 20 publications

Correlation functions with fusion-channel multiplicity in W 3 $$ {\mathcal{W}}_3 $$ Toda field theory

Correlation functions with fusion-channel multiplicity in W 3 $$ {\mathcal{W}}_3 $$ Toda field theory

SU(3) Anderson impurity model: A numerical renormalization group approach exploiting non-Abelian symmetries

Realizing exactly solvableSU(N)magnets with thermal atoms

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