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Modern Physics Letters B

CONTRACTOR RENORMALIZATION GROUP THEORY OF THE EVEN-LEG SPIN TORI

QIHUI CHEN, Peng Li

Published Oct 4, 2010• 0 citations
Publisher

Abstract

The contractor renormalization (CORE) group method is applied to the even-leg spin tori to reveal their ground-state energies and the lowest excitation gaps. The spin tori are referred to the spin ladders that are closed along the rung direction. We designed an improved iterative CORE algorithm and applied it to the even-leg tori. The improvement is made by introducing an extrapolation based on a sequence of hierarchical data that is obtained by varying the size of the elementary block. Quite good results are obtained and compared with the ones by other methods.

DOI
10.1142/s0217984910025036

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The contractor renormalization (CORE) group method is applied to the even-leg spin tori to reveal their ground-state energies and the lowest excitation gaps. The spin tori are referred to the spin ladders that are closed along the rung direction. We designed an improved iterative CORE algorithm and applied it to the even-leg tori. The improvement is made by introducing an extrapolation based on a sequence of hierarchical data that is obtained by varying the size of the elementary block. Quite good results are obtained and compared with the ones by other methods.
The contractor renormalization group (CORE) method is applied to the SU(N)\mathrm{SU}(N)SU(N) chain and ladders in this paper. In our designed schemes, we show that these two classes of systems can return to their original form of Hamiltonian after CORE transformation. Successive iteration of the transformation leads to a fixed point so that the ground state energy and the energy gap to the ground state can be deduced. The result of SU(N)\mathrm{SU}(N)SU(N) chain is compared with the one by Bethe ansatz method. The transformation on spin-1∕21∕21∕2 ladders gives a finite gap in the excited energy spectra to the ground state in an intuitive way. The application to SU(3) ladders is also discussed.
We demonstrate the utility of the numerical contractor renormalization (CORE) method for quantum spin systems by studying one- and two-dimensional model cases. Our approach consists of two steps: (i) building an effective Hamiltonian with longer ranged interactions up to a certain cutoff using the CORE algorithm and (ii) solving this new model numerically on finite clusters by exact diagonalization and performing finite-size extrapolations to obtain results in the thermodynamic limit. This approach, giving complementary information to analytical treatments of the CORE Hamiltonian, can be used as a semiquantitative numerical method. For ladder-type geometries, we explicitely check the accuracy of the effective models by increasing the range of the effective interactions until reaching convergence. Our results in the perturbative regime and also away from it are in good agreement with previously established results. In two dimensions we consider the plaquette lattice and the kagom\'e lattice as nontrivial test cases for the numerical CORE method. As it becomes more difficult to extend the range of the effective interactions in two dimensions, we propose diagnostic tools (such as the density matrix of the local building block) to ascertain the validity of the basis truncation. On the plaquette lattice we have an excellent description of the system in both the disordered and the ordered phases, thereby showing that the CORE method is able to resolve quantum phase transitions. On the kagom\'e lattice we find that the previously proposed twofold degenerate S=1∕2S=1∕2S=1∕2 basis can account for a large number of phenomena of the spin 1∕21∕21∕2 kagom\'e system. For spin 3∕23∕23∕2, however, this basis does not seem to be sufficient. In general we are able to simulate system sizes which correspond to an $8\ifmmode\times\else\texttimes\fi{}8$ lattice for the plaquette lattice or a 48-site kagom\'e lattice, which are beyond the possibilities of a standard exact diagonalization approach.
The COntractor REnormalization group method was devised in 1994 by Morningstar and Weinstein. It was primarily aimed at extracting the physics of lattice quantum field theories (like lattice Quantum Chromodynamics). However, it is a general method of analyzing Hamiltonian lattice systems, e.g. Ising, Heisenberg or Hubbard models. The aim of this work is to show the application of CORE to one-dimensional quantum systems -- the Heisenberg zig-zag model and the Hubbard chain. As a test of the method, the ground state energy of these systems will be calculated.
Contractor renormalization (CORE) is a real-space renormalization-group method to derive effective Hamiltionians for microscopic models. The original CORE method is based on a real-space decomposition of the lattice into small blocks and the effective degrees of freedom on the lattice are tensor products of those on the small blocks. We present an extension of the CORE method that overcomes this restriction. Our generalization allows the application of CORE to derive arbitrary effective models whose Hilbert space is not just a tensor product of local degrees of freedom. The method is especially well suited to search for microscopic models to emulate low-energy exotic models and can guide the design of quantum devices.
We investigate the NNN-leg spin-SSS Heisenberg ladders by using the density matrix renormalization group method. We present estimates of the spin gap ${\ensuremath{\Delta}}_{s}$ and of the ground-state energy per site ${e}_{\ensuremath{\infty}}^{N}$ in the thermodynamic limit for ladders with widths up to six legs and spin $S\ensuremath{\le}\frac{5}{2}$. We also estimate the ground-state energy per site ${e}_{\ensuremath{\infty}}^{2\mathrm{D}}$ for the infinite two-dimensional spin-SSS Heisenberg model. Our results support that for ladders with semi-integer spins the spin excitation is gapless for NNN odd and gapped for NNN even, whereas for integer spin ladders the spin gap is nonzero, independent of the number of legs. Those results agree with the well-known conjectures of Haldane and S\'en\'echal-Sierra for chains and ladders, respectively. We also observe edge states for ladders with NNN odd, similar to what happens in spin chains.
Based on the contractor renormalization group (CORE) method and the density matrix renormalization group (DMRG) method, a new computational scheme, which is called the block density matrix renormalization group with effective interactions (BDMRG-EI), is proposed to deal with the numerical computation of quantum correlated systems. Different from the conventional CORE method in calculating the blocks and the fragments, where the DMRG method instead of the exact diagonalization is employed in BDMRG-EI, BDMRG-EI makes the calculations of larger blocks and fragments applicable. Integrating DMRG's advantage of high accuracy and CORE's advantage of low computational costs, BDMRG-EI can be widely used for the theoretical calculations of the ground state and low-lying excited states of large systems with simple or complicated connectivity. Test calculations on a 240 site one-dimensional chain and a double-layer polyacene oligomer containing 48 hexagons with the spin-1/2 Heisenberg Hamiltonian demonstrate the efficiency and potentiality of the method.