Computational quantum mechanics
The main goal of our activity is to understand the behavior of quantum mechanical many-particle systems from an ab initio perspective, viz. starting from the basic constituents and their mutual interactions. The methods we employ and study cover a broad range of mathematical disciplines, from statistical Monte Carlo methods to large scale diagonalization approaches. The techniques and methods we study are of high relevance to quantum mechanical studies of systems in material science and nanotechnology. The systems we focus on range from Bose-Einstein condensation of atoms, via studies of quantum dots of relevance for nanotechnology to the structure of nuclei and dense matter such as neutron stars.
A theoretical understanding of the behavior of quantum mechanical systems with many interacting particles, normally called many-body systems, is a great challenge and provides fundamental insights into systems governed by quantum mechanics, as well as offering potential areas of industrial applications, from semi-conductor physics to the construction of quantum gates. The capability to simulate quantum mechanical systems with many interacting particles is crucial for advances in such rapidly developing fields like materials science.
However, most quantum mechanical systems of interest in physics consist of a large number of interacting particles. The total number of particles is usually sufficiently large that an exact solution (viz., in closed form) cannot be found. One needs therefore reliable numerical methods for studying quantum mechanical systems with many particles.
Computational quantum mechanics is thus a field of research which deals with the development of stable algorithms and numerical methods for solving Schrödinger's or Dirac's equations for many interacting particles, in order to gain information about a given system. Typical examples of popular many-body methods are coupled-cluster methods, various types of Monte Carlo methods, perturbative expansions, Green's function methods, the density-matrix renormalization group ab initio density functional theory and large-scale diagonalization methods. The numerical algorithms cover a broad range of mathematical methods, from linear algebra problems to Monte Carlo simulations.
Studies of many-body systems span from our understanding of the strong force with quarks and gluons as degrees of freedom, the spectacular macroscopic manifestations of quantal phenomena such as Bose-Einstein condensation with millions of atoms forming a coherent state, to properties of new materials, with electrons as effective degrees of freedom. The length scales range from few micrometers and nanometers, typical scales met in materials science, to ten to the minus eighteen meters, a relevant length scale for the strong interaction. Energies can span from few meV to GeV or even TeV. In some cases the basic interaction between the interacting particles is well-known. A good example is the Coulomb force, familiar from studies of atoms, molecules and condensed matter physics. In other cases, such as for the strong interaction between neutrons and protons (commonly dubbed as nucleons) or dense quantum liquids one has to resort to parameterizations of the underlying interparticle interactions. But the system can also span over much larger dimensions as well, with neutron stars as one of the classical objects. This star is the endpoint of massive stars which have used up their fuel. A neutron star, as its name suggests, is composed mainly of neutrons, with a small fraction of protons and probably quarks in its inner parts. The star is extremely dense and compact, with a radius of approximately 10 km and a mass which is roughly times that of our sun. The quantum mechanical pressure which is set up by the interacting particles counteracts the gravitational forces, hindering thus a gravitational collapse. To describe a neutron star one needs to solve Schrödinger's equation for approximately ten to the 54 interacting particles!
With a given interparticle potential and the kinetic energy of the system, one can in turn define the so-called many-particle Hamiltonian which enters the solution of Schrödinger's equation or Dirac's equation in case relativistic effects need to be included. For many particles, Schrödinger's equation is an integro-differential equation whose complexity increases exponentially with increasing numbers of particles and states that the system can access. Unfortunately, apart from some few analytically solvable problems and one and two-particle systems that can be treated numerically exactly via the solution of sets of partial differential equations, the typical absence of an exactly solvable contribution to the many-particle Hamiltonian means that we need reliable numerical many-body methods. These methods should allow for controlled approximations and provide a computational scheme which accounts for successive many-body corrections in a systematic way.
Due to the huge dimensionalities one encounters in solving many-body problems, methods like partial differential equations can at most be used for 2-3 particles and are of a limited value in computational quantum mechanics, unless one can reduce the problem to an effective one-particle problem. The latter has an applicability only for systems with weak interactions and/or where single-particle degrees of freedom are predominant. Strongly interacting cases such as those found in nuclear physics or quantum liquids have to be treated in full glory.
Our research has focussed mainly on many-body methods applied to problems in nuclear physics, a field where comparison between experiment and exact numerical results, is still lagging behind the precision obtained in fields like quantum chemistry. One major reason for this is the lack of a proper knowledge of the underlying interaction. In nuclear physics, the number of degrees of freedom normally exceed those encountered in quantum chemistry problems. In addition, the interaction itself is not known analytically and three-body forces and correlations play an important role. The latter provide up to 10 per cent or perhaps more of the total binding energy.
The nuclear many-body is perhaps the worst case scenario when it comes to the applicability of many-body methods. To complicate life even more, current experimental programs in low-energy nuclear physics focus on nuclei close to the stability line. This means that one needs to account for the fact that many of the nuclei close to the stability line can be weakly bound and therefore the nuclear interactions will couple bound, continuum, and resonant states. This leads to a further increase in dimensionality compared with stable nuclei.
Many-body theory and requirements to ab initio methods
In order to tackle these challenges, we have chosen to focus on coupled cluster methods in our discussion of systems involving many single-particle degrees of freedom. The ab initio coupled-cluster theory is a particularly promising candidate for such endeavors due to its enormous success in quantum chemistry. Our national quantum chemistry groups, recently awarded grants for a center of excellence on computational chemistry, have been working on coupled cluster methods for more than a decade. Coupled cluster methods results in amazingly precise estimates for systems governed by electrostatic interactions with approximately 100-200 electrons. For larger systems one resorts often to density functional theory, which seldomly goes beyond mean-field approaches (independent particle approximations). The price is often a loss of a quantitative predicability.
Coupled cluster methods allow to study ground- and excited-state properties of systems with dimensionalities beyond the capability of present large-scale diagonalization approaches, with a much smaller numerical effort when compared to diagonalization methods aiming at similar accuracies. Our hope is then that coupled cluster methods, which actually originated from nuclear physics but have found large areas of applications and theoretical developlements in quantum chemistry, can be used to shed light on different many-body correlations in nuclear physics. With a given approach to a many-body Hamiltonian, we hope then to be able to extract enough information to claim that an eventual disagreement with experiment is due to missing physics in the starting Hamiltonian.
The aim of our recent many-body studies is therefore to delineate a many-body scheme for nuclear physics problems based on coupled-cluster theory that incorporates as many as possible of the following features. In addition to coupled-cluster theories, we have also invested quite some efforts on many-body perturbation theory, large-scale diagonalization techniques, Green's function aprpoaches and Variational and diffusion Monte Carlo methods. Together with the coupled-cluster approach, all of the above methods enter our theoretical bag of tools. The theories and methods one uses when studying many-body systems
- should be fully microscopic and start with present two- and three-body interactions derived from, for example, effective field theory or eventually lattice quantum-chromodynamics. The latter is most likely possible by 2013-2015 with the advent of peta-scale computing. One may then have enough statistics to reliably compute various partial wave contributions within the framework of lattice QCD. From lattice QCD calculations, one can constrain the off-shell character of the nucleon-nucleon interaction from the fundamental QCD Lagrangian, removing thereby several ambiguities in the nuclear many-body problem. Presently, we have several methods which allow almost exact solutions of the nuclear many-body problem. The large uncertainty resides in the derivation of the underlying interactions.
- The theory can be improved upon systematically by the inclusion of three-body interactions and more complicated correlations.
- It allows for the description of both closed-shell systems and valence systems.
- It is amenable to parallel computing.
- It can be used to generate excited spectra for nuclei where many shells are involved and describe weakly bound systems with or without resonances and couplings to the continuum.
- Enables the derivation of effective interactions to be used in reduced space appropriate for large-scale diagonalization techniques with matrices with more than a billion eigenstates.
- Enables microscopic nuclear structure results to be married with microscopic reaction studies.
Our research is rooted in the above requirements .
As an example, we have recently performed large-scale Coupled-cluster calculation for Helium isotopes, published recently in Physics Letters B, The calculations were mainly carried out at the Jaguar supercomputing facility at Oak Ridge National Laboratory. The newly upgraded Cray XT4/XT3 supercomputer has a computing power of 101.7 Teraflops, resulting in second place on the top500 computers list of june 2007. In our largest runs we used between 1000 to 2000 nodes.
The coupled-cluster results where obtained at the level of two-particle-two-hole correlations (so-called singles and doubles) for ground state energies of the He isotopes for increasing number of partial waves. In our largest calculation we include the 5s5p5d4f4g4h4i proton orbitals and 20s20p5d4f4g4h4i neutron orbitals, with a complex basis in order to reproduce eventual resonances. In total almost 1000 single particle orbitals where included, Our calculations show excellent convergence with respect to the single-particle basis size. We obtain a convergence within 10 keV for the real part and within 0.1 keV for the imaginary part of the ground state energy. These results here represent the first time that decay widths have been computed in an ab-initio way for an isotopic chain. The decay widths of unbound nuclei are in semiquantitative agreement with experimental data, and the binding energies meet expectations for ab-initio calculations based on two-body Hamiltonians. The calculated masses follow the experimental pattern in that He are unstable with respect to one-neutron emission and He stable with respect to one-neutron emission. The missing agreement with experiment is probably due to the lack of the inclusion of three-nucleon clusters and three-nucleon forces . With the inclusion of the latter we may hopefully be able to tell how much of the spectrum is driven by a coupling to resonances and the non-resonant continuum and how much is due to possible three-nucleon forces. The latter would aid us in explaining one of the major unresolved problems in low-energy nuclear physics, namely how nuclei evolve towards the line of stability.
We are now extending these ab initio to heavier nuclear systems such as tin and nickel isotopes, with mass number A=100 and A=56, respectively.
Finally, the methods we have developed for studies of weakly bound systems can easily be applied to other quantum mechanical systems where resonances play important roles. Quantum dots play an important role in our studies of quantum mechanical systems in solid state physics. In this connection we are also studying the mathematical properties of various many-body methods.
We have also studied the ground states of Bose-Einstein condensates using both variational Monte Carlo and diffusion Monte Carlo methods.
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