Contractor Renormalization Research Articles

P6chiral resonating valence bonds in the kagome antiferromagnet

The Kagome Heisenberg antiferromagnet is mapped onto an effective Hamiltonian on the star superlattice by Contractor Renormalization. Comparison of ground state energies on large lattices to Density Matrix Renormalization Group justifies truncation of effective interactions at range 3. Within our accuracy, magnetic and translational symmetries are not broken (i.e. a spin liquid ground state). However, we discover doublet spectral degeneracies which signal the onset of p6 - chirality symmetry breaking. This is understood by simple mean field analysis. Experimentally, the p6 chiral order parameter should split the optical phonons degeneracy near the zone center. Addition of weak next to nearest neighbor coupling is discussed.

Multigrid Algorithms for Tensor Network States

The widely used density matrix renormalization group (DRMG) method often fails to converge in systems with multiple length scales, such as lattice discretizations of continuum models and dilute or weakly doped lattice models. The local optimization employed by DMRG to optimize the wave function is ineffective in updating large-scale features. Here we present a multigrid algorithm that solves these convergence problems by optimizing the wave function at different spatial resolutions. We demonstrate its effectiveness by simulating bosons in continuous space and study nonadiabaticity when ramping up the amplitude of an optical lattice. The algorithm can be generalized to tensor network methods and combined with the contractor renormalization group method to study dilute and weakly doped lattice models.

Phase separation versus supersolid behavior in frustrated antiferromagnets

We investigate the competition between spin-supersolidity and phase separation in a frustrated spin-half model of weakly coupled dimers. We start by considering systems of hard-core bosons on the square lattice, onto which the low-energy physics of the herein investigated spin model can be mapped, and devise a criterion for gauging the interplay between supersolid order and domain wall formation based on strong coupling arguments. Effective bosonic models for the spin model are derived via the contractor renormalization (CORE) algorithm and we propose to combine a self-consistent cluster mean-field solution with our criterion for the occurrence of phase separation to derive the phase diagram as a function of frustration and magnetic field. In the limit of strong frustration, the model is shown to be unstable toward phase separation, in contradiction with recently published results. However, a region of stable supersolidity is identified for intermediate frustration, in a parameter range not investigated so far and of possible experimental relevance.

CONTRACTOR RENORMALIZATION GROUP THEORY OF THE EVEN-LEG SPIN TORI

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.

Local renormalization method for random systems

In this paper, we introduce a real-space renormalization transformation for random spin systems on two-dimensional (2D) lattices. The general method is formulated for random systems and results by merging two well-known real-space renormalization techniques, namely the strong disorder renormalization technique and the contractor renormalization technique. We analyze the performance of the method on the 2D random transverse field Ising model.

BLOCK DENSITY MATRIX RENORMALIZATION GROUP WITH EFFECTIVE INTERACTIONS

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.

ENCORE: An extended contractor renormalization algorithm

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.

Mechanisms for spin supersolidity inS=12spin-dimer antiferromagnets

Using perturbative expansions and the contractor renormalization (CORE) algorithm, we obtain effective hard-core bosonic Hamiltonians describing the low-energy physics of $S=1/2$ spin-dimer antiferromagnets known to display supersolid phases under an applied magnetic field. The resulting effective models are investigated by means of mean-field analysis and quantum Monte Carlo simulations. A ``leapfrog mechanism,'' through means of which extra singlets delocalize in a checkerboard-solid environment via correlated hoppings, is unveiled that accounts for the supersolid behavior.

Quantum phase transition in a Heisenberg antiferromagnet on a square lattice with strong plaquette interactions

We present numerical results for an $S=1/2$ Heisenberg antiferromagnet on an inhomogeneous square lattice with tunable interaction between spins belonging to different plaquettes. Employing quantum Monte Carlo, we significantly improve on previous results for the critical point separating singlet-disordered and N\'eel-ordered phases and obtain an estimate for the critical exponent $\ensuremath{\nu}$ consistent with the three-dimensional classical Heisenberg universality class. Additionally, we show that a fairly accurate result for the critical point can be obtained from a contractor renormalization expansion by applying a surprisingly simple analysis to the effective Hamiltonian.

Engineering exotic phases for topologically protected quantum computation by emulating quantum dimer models

We use a nonperturbative extended contractor renormalization (ENCORE) method for engineering quantum devices for the implementation of topologically protected quantum bits described by an effective quantum dimer model on the triangular lattice. By tuning the couplings of the device, topological protection might be achieved if the ratio between effective two-dimer interactions and flip amplitudes lies in the liquid phase of the phase diagram of the quantum dimer model. For a proposal based on a quantum Josephson junction array [L. B. Ioffe , Nature (London) 415, 503 (2002)] our results show that optimal operational temperatures below 1 mK can only be obtained if extra interactions and dimer flips, which are not present in the standard quantum dimer model and involve three or four dimers, are included. It is unclear if these extra terms in the quantum dimer Hamiltonian destroy the liquid phase needed for quantum computation. Minimizing the effects of multidimer terms would require energy scales in the nano-Kelvin regime. An alternative implementation based on cold atomic or molecular gases loaded into optical lattices is also discussed, and it is shown that the small energy scales involved-implying long operational times-make such a device impractical. Given the many orders of magnitude between bare couplings in devices, and the topological gap, the realization of topological phases in quantum devices requires careful engineering and large bare interaction scales.

Bootstrap approximations in contractor renormalization

We propose a bootstrap method for approximating the long-range terms in the contractor renormalization method. The idea is tested on the two-dimensional Heisenberg antiferromagnet and the frustrated ${J}_{2}\text{\ensuremath{-}}{J}_{1}$ model. We obtain renormalization group flows that directly reveal the N\'eel phase of the unfrustrated Heisenberg antiferromagnet and the existence of a phase transition in the ${J}_{2}\text{\ensuremath{-}}{J}_{1}$ model for weak frustration. However, we find that this bootstrap method is dependent on blocking and truncation schemes. For this reason, we discuss these dependencies and unresolved issues that researchers who use this approach must consider.

Contractor-renormalization approach to frustrated magnets in a magnetic field

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 non-perturbative 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 usual perturbation theory. We also address the issue of plateau formation in two-dimensional bilayers and point out the analogy between non-frustrated strongly anisotropic models and frustrated SU(2) ones.

Exploring contractor renormalization: Perspectives and tests on the two-dimensional Heisenberg antiferromagnet

Contractor Renormalization (CORE) is a numerical renormalization method for Hamiltonian systems that has found applications in particle and condensed matter physics. There have been few studies, however, on further understanding of what exactly it does and its convergence properties. The current work has two main objectives. First, we wish to investigate the convergence of the cluster expansion for a two-dimensional Heisenberg Antiferromagnet(HAF). This is important because the linked cluster expansion used to evaluate this formula non-perturbatively is not controlled by a small parameter. Here we present a study of three different blocking schemes which reveals some surprises and in particular, leads us to suggest a scheme for defining successive terms in the cluster expansion. Our second goal is to present some new perspectives on CORE in light of recent developments to make it accessible to more researchers, including those in Quantum Information Science. We make some comparison to entanglement-based approaches and discuss how it may be possible to improve or generalize the method.

Application of Contractor Renormalization Group to the Heisenberg Zig-Zag and the Hubbard Chain

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 contractor renormalization group method 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.

Effective Hamiltonian Approach for Strongly Correlated Systems

We demonstrate the utility of effective Hamilonians for studying extended 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 explicitly 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 kagome 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 kagome 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 kagome system. In general, we are able to simulate system sizes which correspond to an 8× 8 lattice for the plaquette lattice or a 48-site kagome lattice, which are beyond the possibilities of a standard exact diagonalization approach.

Contractor renormalization group theory ofSU(N)chains and ladders

Contractor renormalization group (CORE) method is applied to the 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($N$) chain is compared with the one by Bethe ansatz method. The transformation on spin-1/2 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.

Low-Energy Singlets in the Heisenberg Antiferromagnet on the Kagome Lattice

The spin-1/2 Heisenberg antiferromagnet on the kagomé lattice, is mapped by contractor renormalization to a spin-pseudospin Hamiltonian on the triangular superlattice. Variationally, we find a ground state with columnar dimer order. Dimer orientation fluctuations are described by an effective O(2) model at energies above an exponentially suppressed clock mass scale. Our results explain the large density of low-energy singlets observed numerically, and the nonmagnetic T2 specific heat observed experimentally.

Numerical contractor renormalization method for quantum spin models

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 $S=1∕2$ basis can account for a large number of phenomena of the spin $1∕2$ kagom\'e system. For spin $3∕2$, 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.

Superconducting gap for a two-leg t-J ladder.

Single-particle diagonal and off-diagonal Green's functions of a two-leg t-J ladder at 1/8 doping are investigated by exact diagonalizations techniques. A numerically tractable expression for the superconducting gap is proposed and the frequency dependence of the real and imaginary parts of the gap are determined. The role of the low-energy gapped spin modes, whose energies are computed by a (one-step) contractor renormalization procedure, is discussed.

Singlet Excitations in Pyrochlore: A Study of Quantum Frustration

We apply the contractor renormalization (CORE) method to the spin half Heisenberg antiferromagnet on the frustrated checkerboard and pyrochlore lattices. Their ground states are spin-gapped singlets which break lattice symmetry. Their effective Hamiltonians describe fluctuations of orthogonal singlet pairs on tetrahedral blocks, at an emergent low energy scale. We discuss low temperature thermodynamics and new interpretations of finite size numerical data. We argue that our results are common to many models of quantum frustration.