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arXiv:cond-mat/0510785 · AIP conference proceedings

Numerical Contractor Renormalization applied to strongly correlated systems

Sylvain Capponi

Published Jan 1, 2006• 1 citation
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Abstract

We demonstrate the utility of effective Hamilonians for studying strongly correlated systems, such as quantum spin systems. After defining local relevant degrees of freedom, the numerical Contractor Renormalization (CORE) method is applied in two steps: (i) building an effective Hamiltonian with longer ranged interactions up to a certain cut‐off 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 semi‐quantitative 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 both in the doped and undoped case are in good agreement with previously established results. In two dimensions we consider the plaquette lattice and the kagomé lattice as non‐trivial 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 kagomé lattice we find that the previously proposed twofold degenerate S = 1/2 basis can account for a large number of phenomena of the spin 1/2 kagomé system and gives a good starting point to study the doped case.

DOI
10.1063/1.2178028

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We demonstrate the utility of effective Hamilonians for studying strongly correlated systems, such as quantum spin systems. After defining local relevant degrees of freedom, the numerical Contractor Renormalization (CORE) method is applied in two steps: (i) building an effective Hamiltonian with longer ranged interactions up to a certain cut‐off 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 semi‐quantitative 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 both in the doped and undoped case are in good agreement with previously established results. In two dimensions we consider the plaquette lattice and the kagomé lattice as non‐trivial 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 kagomé lattice we find that the previously proposed twofold degenerate S = 1/2 basis can account for a large number of phenomena of the spin 1/2 kagomé system and gives a good starting point to study the doped case.
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.
A review of the Contractor Renormalization (CORE) method, as a systematic derivation of the low energy effective hamiltonian, is given, with emphasis on its differences and advantages over traditional perturbative (weak/strong links) real space RG. For the low energy physics of the 2D Hubbard model, we derive the plaquette bosons (projected SO(5)) model which connects the microscopic model to phases and phenomenology of high‐Tc cuprates. For the S = 1/2 Pyrochlore and Kagomé antiferromagnets, the effective hamiltonians predict spin‐disordered, lattice symmetry breaking, ground states with a large density of low energy singlets as found by exact diagonalization of small clusters.
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.
We propose to use the contractor-renormalization (CORE) technique in order to derive effective models for quantum magnets in a magnetic field. CORE is a powerful nonperturbative technique that can reduce the complexity of a given microscopic model by focusing on the low-energy part. We provide a detailed analysis of frustrated spin ladders, which have been widely studied in the past: in particular, we discuss how to choose the building block and emphasize the use of their reduced density matrix. With a good choice of basis, CORE is able to reproduce the existence or not of magnetization plateaux in the whole phase diagram contrary to a usual perturbation theory. We also address the issue of plateau formation in two-dimensional bilayers and point out the analogy between nonfrustrated strongly anisotropic models and frustrated SU(2) ones.
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.
The contractor renormalization group (CORE) method, a new approach to solving Hamiltonian lattice systems, is introduced. The method combines contraction and variational techniques with the real-space renormalization group approach. It applies to lattice systems of infinite extent and is ideal for studying phase structure and critical phenomena. The CORE approximation is systematically improvable and can treat systems with dynamical fermions. The method is tested using the (1+1)-dimensional Ising model.