Fernando Arias de Saavedra
Universidad de Granada

Joaquin E. Drut
University of North Carolina

Jonathan L. DuBois
Lawrence Livermore National Laboratory

Stefano Gandolfi
Los Alamos National Laboratory

Francesco Pederiva
University of Trento - INFN

Program Coordinator:
Laura Lee
(206) 685-3509

Seminar Schedules:

  • Week 1 (June 24-28)
  • Week 2 (July 1-5)
  • Week 3 (July 8-12)
  • Week 4 (July 15-19)
  • Week 5 (July 22-26)
  • Week 6 (July 29-aug2)

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    INT Program INT-13-2a

    Advances in quantum Monte Carlo techniques for
    non-relativistic many-body systems

    June 24 - August 2, 2013


    The use of Quantum Monte Carlo (QMC) methods has now become ubiquitous in many- body physics. Present applications cover essentially the whole spectrum of non-relativistic physics:
    • Low-energy nuclear structure and nuclear reactions;
    • Physics of Bosons and Fermions at ultra-low temperatures;
    • Quantum fluids and solids (4He, 3He, H2);
    • Electronic structure of atoms and molecules;
    • Solid state physics and materials science.
    Part of the success of the method is due to the increased availability of high-performance parallel computing resources, to which QMC algorithms are easily adapted. There are two different families of QMC algorithms for solving the Schrödinger equation at T=0.
    • Variational Monte Carlo calculations, in which a parametrized ansatz for the solution of a many-body Schrödinger equation is made. The wavefunction is then optimized exploiting some variational principle (usually on the expectation value of the Hamiltonian or on its variance). This approach is of fundamental importance as it provides the ingredients to more sophisticated calculations. However, in some cases, like the study of quantum systems in phase coexistence, it is the only viable approach.
    • Imaginary-time propagation methods, i.e. methods based on the fact that, [Equation here] converges to the absolute ground state of a given Hamiltonian H for [tau to infinity]. This goal can be reached by propagating a set of points in configuration space (Green’s Function MC, Diffusion MC), or by path-based methods (Path-Integral Ground State MC, Reptation MC). This set of algorithms is commonly viewed as a way to solve “exactly” the quantum many-body problem. However, it is also well known that when applied to many-fermion systems, a set of approximations must be used to avoid the so-called “sign problem”. For Hamiltonians including an operator- dependent potential (as in the case of realistic nucleon-nucleon interactions or spin-orbit for electrons) it is also convenient to introduce sets of auxiliary variables (Auxiliary Fields QMC).

    It has also been possible to extend this second set of algorithms to the study of Effective Field Theories (EFTs) for many-nucleon systems. More recently, methods based on determining the occupation of states of a basis set (in the spirit of more traditional Quantum Chemistry calculations) have been introduced. Lattice QMC methods, which involve single particle orbitals defined on a discretization of coordinate space, have been in use for a long time. In general, these methods do not display a sign problem in the special case of a purely attractive interaction, as long as there is no spin polarization. Very recently, it was shown that general Hamiltonians can be studied on the lattice at T=0 without a sign problem, but careful investigations of the signal-to-noise properties of the operators of interest are required.

    At finite temperature, it is possible to use path-based methods, which go under the general name of “Path Integral MC”, aiming to sample the density matrix of a quantum system at finite temperature. This technique becomes extremely useful to study the transition to quantum condensate phases and in general the onset of coherence phenomena. Also at finite temperature it is possible to use Lattice MC methods in the grand-canonical ensemble, with periodic (bosons) or anti-periodic (fermions) boundary conditions in the imaginary-time direction using Hubbard-Stratonovich fields to represent the interactions. The concept of imaginary-time propagation has also been recently exploited to extract information about excited states and the response function of a many-body system. This problem is obviously related to the development of methods to accurately perform the imaginary time to frequency transformation.

    Both at zero and non-zero temperature, it has been shown that methods imported from the area of Lattice QCD can aid in the development of more efficient techniques for non- relativistic systems. This is true in particular of Hybrid Monte Carlo (HMC) methods, without which Lattice QCD would not have achieved its current status. Aside from a few exceptions, HMC has not been widely applied outside Lattice QCD yet.

    In contrast to the steady development and wide use by non-experts of Density Functional Theory based methods, QMC yet to find a stable working standard. Recently, some packages have been released that allow to perform calculations for materials science and quantum chemistry problems using the code as a “black box”. However, in most cases users of the method are also developers, and most codes are written ad hoc for a given problem. The diversification of the subjects disfavored the percolation of technical advancements across the communities. For instance, refined wavefunction optimization is a considerable issue in quantum chemistry calculations, it is underutilized in simulations of quantum fluids and solids or for certain materials science calculations, and is usually almost disregarded in nuclear physics calculations. On the contrary, all the technology for dealing with complex wavefunctions is necessary in nuclear physics calculations, while in condensed matter problems almost no attention is payed to this field -- despite the fact that Hamiltonians breaking the time-reversal symmetry are of significant physical interest.

    We find it necessary, in this context, to set up an occasion for scientists from the three communities of quantum chemistry, condensed matter physics, and nuclear physics to gather and exchange expertise and progress in all the aspects of Quantum Monte Carlo techniques for continuum systems that go beyond the mere application of standard algorithms. In particular, our main objectives include:

    • Understanding the level of progress that has been already reached, and that is reachable in the future, towards controllable accuracy in many-fermion calculations. This implies the building of a comprehensive understanding of all the approximations (the standard fixed-node, but also the fixed-phase approximation for the case of DMC calculations, constrained-path approximations for path-based algorithms and systematic uncertainties in lattice methods) as well as the possible ways of going beyond them, by means of more efficient transient calculations.
    • Promoting a definition of a standard level of accuracy of QMC calculations that can be applied across different research fields.
    • Defining a meaningful benchmark problem to be used in order to systematically compare the different methods and approximations, both for ground state and finite temperature calculations. We expect the experience of the recent workshop INT-11-1, Fermions from Cold Atoms to Neutron Stars: Benchmarking the Many-Body Problem, March 14- May 20, 2011 to prove useful for this purpose.

    The proposed program bears a strong connection to other recent INT programs, such as INT-11-1 (mentioned above), INT-11-2a (Extreme computing), INT-10-1 (Simulations and symmetries) and INT-09-1 (EFT & the many-body problem). In this sense, we expect this program to play an important role in revitalizing the communication of methods across fields, as well as maintaining the continued discussion and interest in the development of stochastic methods for the many-body problem within nuclear physics and beyond.