Graphical Methods for Spin Systems

L. Chayes, Jonathan L. Machta

Khazret S. Nirov, A. V. Razumov

We systematise and develop a graphical approach to the investigations of quantum integrable vertex statistical models and the corresponding quantum spin chains.The graphical forms of the unitarity and various crossing relations are introduced.Their explicit analytical forms for the case of integrable systems associated with the quantum loop algebra U q (L(sl l+1 )) are given.The commutativity conditions for the transfer operators of lattices with a boundary are derived by the graphical method.Our consideration reveals useful advantages of the graphical approach for certain problems in the theory of quantum integrable systems.

Antônio Sérgio Teixeira Pires

Werner Dobrautz, Simon D. Smart, Ali Alavi

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.