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 and the recent extension of
the so called ''fixed node approximation'' (FN) to lattice models,
it has been recently possible to define a novel variational approach (SR)
with the two important properties of a systematic and accurate computational
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 electrons.
Hereafter we mention the most important numerical achievements obtained
1) D-wave superconductivity of the t-J model.
2) RVB-spin-liquid wavefunction in a frustrated Heisenberg model.
3) Ferromagnetism in the infinite-U Hubbard model.
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.
 E, S, Heeb and T. M. Rice, Europhys. Lett. 82, 3899 (1994).
 D, F, B. ten Haaf et al., Phys. Rev. B 51, 13039 (1995).
 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).
 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).
 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).
 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
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 , with and increasing computational time , 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 , not exceedingly small. Preliminary test cases on simple atoms suggest
that an appropriate lattice regularization, defined to be exact (with
no dependence) for the Hartee-Fock approximate wavefunction, may fulfill
the previous requirement ( with
for the exact ground state calculation), even for
atoms with large electron number .
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 email@example.com.
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
CURRENT RESEARCH TOPICS IN STRONGLY CORRELATED ELECTRON
1) Numerical study and modeling of Mott insulators with unconventional
[S. SORELLA in collaboration with L. Arrachea, F. Becca, L. Capriotti,
A. Parola (Como) and G. Santoro.] Main references: , , , 
, 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 , 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
[S. SORELLA in collaboration with V. Anisimov (Ekatherinburg, Russia),
E. Dagotto (Tallahassee USA), A. Parola, F. Becca, and S. Yunoki.]
Main references: ,,, 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
, by a newly developed quantum Monte Carlo scheme , 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.