QUANTUM MONTE CARLO FOR STRONGLY CORRELATED ELECTRONS

In the recent years Quantum Monte Carlo (QMC) methods for lattice models have been considerably improved. By using the systematic convergence of the Lanczos method to the ground state[1] and the recent extension of the so called ''fixed node approximation'' (FN) to lattice models[2], it has been recently possible to define a novel variational approach (SR) with the two important properties of a systematic and accurate computational technique[3]:

I) Systematic convergence to the exact solution:
as for the Lanczos technique, the SR-method converges to the exact ground state of a lattice model for ''large enough'' computer time. This important property is certainly missing for most approximate numerical techniques. For instance, within approximate DFT schemes, a large degree of arbitrariness is given by the choice of pseudo-potentials and/or types of gradient corrections to LDA. Within the SR or Lanczos techniques there exist instead a unique way to improve the calculation. This is obtained just by increasing the number of iterations, obviously with larger (or much larger for large number of electrons) computational effort.

II) Accuracy in correlation functions:
At each iteration the approximate variational wavefunction is the ground state of a physical Hamiltonian which is as close as possible to the exact Hamiltonian. This property is not satisfied by the Lanczos technique but is guaranteed within FN (that however cannot satisfy property I) or SR schemes. In this way ground state correlation functions converge much faster to the exact ones and in many cases ''large enough'' computer time becomes possible with present PC's even for $~100$ electrons.

Hereafter we mention the most important numerical achievements obtained recently:

1) D-wave superconductivity of the t-J model[4].
2) RVB-spin-liquid wavefunction in a frustrated Heisenberg model[5].
3) Ferromagnetism in the infinite-U Hubbard model[6].

All these topics have been intensively debated in the literature for at least a decade and have been ''solved numerically'' (i.e. within the limitation of a finite size calculation) by the present approach.

[1] E, S, Heeb and T. M. Rice, Europhys. Lett. 82, 3899 (1994).

[2] D, F, B. ten Haaf et al., Phys. Rev. B 51, 13039 (1995).

[3] S. Sorella, ''Green Function Monte Carlo with Stochastic Reconfiguration'' Phys. Rev. Letters 80, 4558 (1998); ibidem ''Generalized Lanczos algorithm for Variational Monte Carlo'' Phys. Rev. B 64, 024512 (2001).

[4] S. Sorella, G. Martins, F. Becca, C. Gazza, L. Capriotti, A. Parola and E. Dagotto, '' Superconductivity in the 2D t-J model'', Phys. Rev. Lett. 88 ,117002 (2002).

[5] L. Capriotti, Federico Becca, Alberto Parola and Sandro Sorella, ''Resonating Valence Bond Wave Functions for Strongly Frustrated Spin Systems'' Phys. Rev. Lett. 87, 097201 (2001).

[6] Federico Becca, Sandro Sorella ''Nagaoka Ferromagnetism in the 2D- infinite U Hubbard model'', Phys. Rev. Lett. 86, 3396 (2001). FUTURE

 

PERSPECTIVES Lattice regularization of the first-principle all electron Hamiltonian
Our main project in the recent months is to define a computational scheme that satisfies properties (I) and (II) for the realistic Hamiltonian with electron-electron Coulomb interaction, within the Born-Oppenheimer approximation. In principle it is possible to avoid the problem of pseudopotentials and correction schemes to LDA, by defining the realistic Hamiltonian on a lattice with a fixed mesh size $a$, with $a \to 0$ and increasing computational time $\simeq 1/a^2$, consistent with the requirement (I).
''Large enough'' computer time to obtain a reasonable accuracy in correlation functions (II)- in particular the Born-Oppenheimer forces acting on the nuclei- becomes possible, provided the lattice regularization of the continuous problem is appropriate, namely allowing a lattice mesh $a$, not exceedingly small. Preliminary test cases on simple atoms suggest that an appropriate lattice regularization, defined to be exact (with no $a$ dependence) for the Hartee-Fock approximate wavefunction, may fulfill the previous requirement ( with $a \simeq Z^{-1/3}$ for the exact ground state calculation), even for atoms with large electron number $Z$.
In this way a reasonable mesh size may be used for an acceptable physical and chemical accuracy, within a computationally possible ''all electron'' calculation. With the proposed approach all the forces acting on the nuclei -within Born-Oppenheimer approximation- can be computed efficiently with QMC schemes, and also long range forces -such as the so important Van der Waals ones- are consistently included. This project requires many human resources for its practical implementation to realistic materials. However it is important to devote much effort to this project, considering its possible applicability to a vast range of materials not yet understood within LDA. For instance Fe2+,or Fe3+ ions in biological environment are so important for enzymatic catalysis but LDA is not even able to reproduce their stable spin configurations. At the moment I am working alone on this project (no students and no postdocs), that however I feel extremely important. If you wish to speed up the future possibilities of QMC schemes for realistic calculations, by helping me with a motivated and qualified collaboration, please do not hesitate to contact me by sorella@sissa.it.

LIST OF PUBLICATIONS.
Most of my publications (over 70) are in the APS Journals (Phys. Rev. B and Phys. Rev. Letters) and can be easily retrieved in the web site: LINK www.aps.org. For the most recent preprints, please consult the e-print archive: http://babbage.sissa.it/archive/cond-mat

 

CURRENT RESEARCH TOPICS IN STRONGLY CORRELATED ELECTRON SYSTEMS

1) Numerical study and modeling of Mott insulators with unconventional magnetic properties.
 [S. SORELLA in collaboration with L. Arrachea, F. Becca, L. Capriotti, A. Parola (Como) and G. Santoro.] Main references: [2], [14], [37], [54] ,[135] in http://www.sissa.it/ tartagli/cmpapers.html. One of the most important theoretical problems in the theory of Mott insulators is whether they can be adiabatically connected to band insulators, their nature being simple and well understood, or, on the contrary, whether the Mott insulator may characterize a genuine new state where correlation plays a crucial role. For instance, within the latter more appropriate definition, the 2D Hubbard model at one electron per site filling can be indeed considered as a conventional band insulator, because the antiferromagnetic order parameter-in a simple Hartree-Fock picture- reduces the Bruillouin zone volume and allows a conventional band-like explanation of its insulating properties. Recently ] we have shown numerically [37,135] that in a two dimensional (2D) frustrated spin model it is possible to stabilize a spin liquid ground state with no conventional order parameter, suggesting that the second more exciting possibility- consistent with the resonating valence bond (RVB) theories- is actually still open and alive. The existence of a disordered spin-liquid state is in fact the necessary condition for stabilizing a correlated Mott insulator. We argue that with the RVB wavefunctions defined in [135], and with recent extensions, it appears possible to understand most frustrated quantum spin 1/2 hamiltonians in low dimensional systems. Work is in progress on several other models ( triangular and Kagome' lattices, zig-zag ladders, frustrated one dimensional chains, and so on and so forth).

2) Relevance of single band models such as t-J, Hubbard, with or without next-nearest neighbor hoppings, in the context of High Temperature superconductivity.
[S. SORELLA in collaboration with V. Anisimov (Ekatherinburg, Russia), E. Dagotto (Tallahassee USA), A. Parola, F. Becca, and S. Yunoki.]
Main references: [69],[115],[137],[161] in http://www.sissa.it/ tartagli/cmpapers.html.

The mechanism of High-temperature superconductivity (HTc) remains until now an highly debated issue. Our group has contributed to the scientific discussion (or at least to remove some prejudices) by finding clear numerical support to the original idea pointed out by P. W. Anderson at the early stages of HTc. According to this theory (Science 1987) a single-band model, in presence of strong correlation, can explain not only High-temperature superconductivity but also all the anomalous properties (linear-T resistivity, pseudo-gap behavior, anomalous photoemission spectra) measured in these materials and still unexplained with conventional theories. The basic point of the theory is that the ground state of the low energy effective one-band model hamiltonian (t-J or Hubbard at strong coupling) at zero hole-foping is very well described by a RVB wavefunction, where preformed Cooper pairs with d-wave symmetry are necessary to describe the singlet valence bonds of the RVB. In this way it is very natural to expect that, upon small doping, the experimentally measured d-wave superconductivity (forbidden by the constraint of no doubly-occupancy at zero hole-doping) can be established in the t-J model. This has been recently verified numerically [161], by a newly developed quantum Monte Carlo scheme [125], that so far represents the most accurate numerical technique for two dimensional strongly ] correlated electrons. Though our work certainly is not the definite answer to High-Tc and several questions remain still open, it represents a promising line of theoretical and numerical investigation that may allow to understand completely these important phenomena. Recently in collaboration with V. Anisimov we have also been able to reproduce many experimental aspects (antiferromagnetic and superconducting zero temperature transition) of the phase diagram of the most popular HTc compound La_2 Cu O_4 doped with Sr or Nd, with an ` ab-initio calculation (the effective J and long range hoppings were computed within LDA+U calculations) that neglets only the electron-phonon coupling. The inclusion of this interaction may stabilize true static stripes at commensurate fillings (e.g. doping 1/8), and is being currently investigated.