Density Matrix Renormalisation Group Studies of Quantum Lattice Models

           

 

Steven White's density matrix renormalisation group (DMRG), developed in 1993, has revolutionised computational condensed matter physics and is set to have a strong impact upon the solution of other quantum many body problems, where one must take into account interactions between many particles treated at a quantum mechanical level. Examples include quantum chemistry, atomic physics and lattice gauge theory. In this project we have been developing and applying some state-of-the-art DMRG algorithms to models of spin-Peierls compounds and conjugated polymers.

Spin-Peierls compounds are of interest due to the recent discovery that the inorganic compound CuGeO3 undergoes a spin-Peierls transition. The compound consists of weakly coupled chains of antiferromagnetically interacting spins (i.e. the magnetic properties are quasi-one-dimensional). The spins are coupled to the lattice degrees of freedom (ions, or phonons) in that the antiferromagnetic interaction (superexchange) between neighbouring spins depends on the separation of the Cu ions on which they reside. At a temperature of 14K, a transition occurs whereby the lattice dimerises i.e. the lattice orders into a state with alternating long and short bonds between successive Cu ions. The cost in elastic energy of this lattice buckling is compensated by a gain in magnetic energy (the spins like to bind in pairs). The spin-Peierls effect was seen in the 1970s in organic compounds but only after the discovery of CuGeO3 have experimentalists been able to use sensitive probes like neutron scattering, NMR, ESR and a myriad of other techniques to determine quantities such as the magnetic excitation spectrum. The reason for the enormous level of interest in the material since this discovery in 1993 is that electron-phonon interactions play a key role in processes such as high temperature superconductivity and CuGeO3 affords physicists a chance to critically assess theories of the electron-phonon interaction under controlled conditions.

The discovery of electroluminescence in the poly(para-

 

Principal Investigator

Robert J. Bursill

School of Physics

University of New South Wales

Co-Investigators

C. J. Hamer

P. Sriganesh

School of Physics

University of New South Wales

Projects

h04 - VPP, MDSS

   
         
           
- Appendix A

           

       

phenylenevinylene) (PPV) has lead to a huge surge in experimental and theoretical interest in conjugated polymers due to the consequent development commercial devices such as light emitting diodes, large flexible displays, optical switches and light harvesting arrays.

What are the results to date and the future of the work?

Four-block DMRG methods for electron-phonon problems.

A new DMRG technique we have developed, the four-block DMRG method, has been shown to be capable of accurately determining properties of the demanding models that describe these electron-phonon systems on chains of hundreds of sites whereas previous studies were restricted to a handful of lattice sites and were thus inconclusive. In this project we have been exploiting the significant competitive advantage offered by this technique to solve some hither- to intractable problems in the modelling of electron-phonon systems in general and CuGeO3 in particular. This project commenced in 1997-8 (in 1997 some CPU time from project g57 was used) with the vectorisation of the four-block DMRG code we developed on the Power Challenge machine at NSWCPC. To the best of our knowledge this is the only example of a vectorisable DMRG algorithm and gives us an advantage over other DMRG groups in that vector supercomputing power can be applied to solve the models.

The first application of the method was to the spinless fermion Holstein model. This models the Peierls transition from metal to insulator that occurs in quasi-1D molecular crystals. Through our calculations we were able to determine the phase boundary and the exponents which describe the critical phenomena in this model with high accuracy whereas previous studies were inconclusive.

We also performed and published some benchmarks to assess the convergence of the algorithm by making comparisons with exact results for exactly solvable models.

In the second half of 1998 we moved onto the Peierls-Heisenberg model of spins interacting with phonons (the lattice). This should be a reasonable model for the chains in CuGeO3. Most theoretical approaches to CuGeO3 to date have been based on the assumption that the parameter regime is such that the phonons can be treated classically (i.e. the characteristic vibrational frequency of the lattice is a lot smaller than the typical magnetic energy scale) and/or that there is little difference, qualitatively or quantitatively, between treating the phonons classically and treating them fully quantum mechanically. Using the four-block DMRG method we determined the phase diagram and critical behaviour of this model with high accuracy . We showed that both of the above assumptions are in fact incorrect, not only for CuGeO3, but for all spin-Peierls materials where the relevant model parameters have been experimentally determined.

In 1999 we plan to analyse and interpret the plethora of available experimental data for CuGeO3 within the Peierls-Heisenberg model with the phonons treated fully quantum mechanically. We will then extend the model to include magnetic field terms and interchain interactions to interpret field and pressure experiments on CuGeO3. Because of the competitive advantage offered by our code, we will also be in a unique position to study a recent series of experiments on other spin-Peierls materials.

       
Appendix A -

       

     

Semi-empirical models of conjugated polymers.

Conjugated polymers have unique and useful electronic and optical properties which are largely determined by the backbone of p-conjugated carbon atoms. In the simplest of the semi-empirical theories, the Pariser-Parr-Pople (P-P-P) theory, each conjugated carbon atom is represented by a Hubbard site i.e. a single orbital which can be occupied by one or two valence electrons. The model has two parameters which set the strength of the long range Coulomb repulsion between electrons and the energy associated with hopping between two bonded carbon sites. These parameters can be determined by fitting results for excitation energies to small molecules such as benzene, for which accurate experimental data is available. For the molecules that have been studied it has been shown that the P-P-P theory, when solved without any approximations for the electron correlations, is remarkably accurate at predicting excitation energies. Due to the fact that the size of the Hilbert space grows exponentially with the number of carbon atoms, until very recently, it was not possible to diagonalise the P-P-P model Hamiltonian for molecules with more than around 14 carbon atoms.

The advent of the DMRG gives us the means to calculate accurate excitation energies for large systems within the full P-P-P theory (with no approximations for the electron correlations) although the task is not trivial. Along with Dr Barford from The Centre for Molecular Materials at the University of Sheffield, Bursill has developed a general code for solving conjugated polymers. The first application, to polyacetylene (PA), the simplest of the conjugated polymers, produced accurate energies for chains with hundreds of carbon atoms. In this study we modelled the adiabatic relaxation of the lattice (deformation of the backbone upon excitation) and hence the geometries for various excited states. In the process we uncovered a number of interesting new results concerning the soliton structures of the lattice and the magnitude of the relaxation energies which were not apparent from approximate studies. Code has now been developed to study the more demanding and commercially more important phenyl-based polymers such as PPV. This work will carry through into 1999 and beyond. We also plan to develop code to study the effect of dynamical phonons (vibronic effects) and interchain interactions (solvent effects) in PA.

Four-block algorithm for two dimensionsional quantum lattice models

As mentioned, the DMRG has revolutionised computational condensed matter physics. However, the exquisite accuracy achieved by the method in 1D has not been replicated in 2D where the method is merely competitive with the preeminent technique (quantum Monte Carlo (QMC) ) . Later in 1999, we hope to extend the four-block algorithm to 2D quantum lattice systems. We believe that the four-block scheme has some distinct advantages over conventional 2D DMRG algorithms which can be exploited to yield new levels of accuracy. We will try this approach for the 2D Heisenberg model, which describes the antiferromagnetic precursor materials that are doped to form high temperature superconductors. For this model we can compare our results with accurate QMC data. We will then move onto frustrated models, where two types of antiferromagnetic interaction compete, attempting to impose opposing types of magnetic ordering. We will be able to investigate the nature of the controversial spin liquid phase of this model which has been inaccessible to QMC techniques as a result of the infamous minus sign problem which plagues QMC methods when applied to doped or frustrated systems.

     
- Appendix A

 
     

       

What computational techniques are used?

The DMRG technique is a truncated basis expansion for finding a number of low energy eigenstates of quantum lattice Hamiltonians (energy operators) on lattices of increasing size. The advantage of the technique is that it keeps the size of the truncated Hilbert space fixed as the lattice is grown while retaining excellent accuracy for the eigenvalues and wavefunctions, e.g., for the spin-1 antiferromagnetic chain, the infinite lattice ground state energy density was found to machine precision. This is to be compared with straightforward exact diagonalisation for which the Hilbert space grows exponentially with the size of the lattice and hence hits a brick wall at around 16-30 lattice sites. The four-block DMRG technique is particularly well suited to systems with a large number of degrees of freedom per site, electron-phonon models being a prime example. Implementing the method involves a lot of linear algebra-representation and rotation of operators, dense matrix diagonalisation of large matrices (of order 2500-10000) and sparse matrix diagonalisation of very large matrices (of order 15000000) using the conjugate gradient technique. The four-block technique is the first example of a DMRG algorithm with a natural vectorisability. This gives our group a significant competitive advantage in that we can use vector machines such as the VPP at ANUSF to perform the world's largest DMRG calculations. Average vectorisation levels are around 70% with the largest codes achieving around 85%. The single processor speedup from the SGI Power Challenge (PC) to the VPP is around 70 for problems that could be run on the PC. The large vector registers play and important role here as many large arrays are accessed by the inner loops.

The P-P-P theory DMRG codes so far have not been amenable to vectorisation though the VPP has been used to perform some benchmark exact diagonalisations of very large systems (14 carbon atoms) which were used to assess the DMRG convergence. These exact diagonalisation algorithms are highly vectorisable and involve scatter summing, i.e., indirect, random access of very large arrays (around 80 MB). Again the speedup in going from superscalar to vector architecture is of the order of 70 and for both the four-block DMRG and exact diagonalisation codes use is being made of the 2GB processors. For the exact diagonalisation codes large operator matrix elements that are repeatedly used are stored on and retrieved from the MDSS.

Publications

R. J. Bursill, R. H. McKenzie and C. J. Hamer, Density matrix renormalisation group study of the one-dimensional Holstein model of spinless fermions, Physical Review Letters 80, 5607 (1998).

R. J. Bursill, 1999, A density matrix renormalisation group method for quantum lattice systems with a large number of states per site, to appear in Physical Review B.

R. J. Bursill and W. Barford, Electron-lattice relaxation, and soliton structures and their interactions in polyenes, Physical Review Letters 82, 1514 (1999).

R. J. Bursill, R. H. McKenzie, and C. J. Hamer, 1999, Phase diagram of a Heisenberg spin-Peierls model with quantum phonons, submitted to Phys. Rev. Lett.

R. J. Bursill C. Castleton and W. Barford, Chem. Phys. Lett. 294, 305 (1998).

       
Appendix A -