Lanczos approach using out-of-core memory for eigenvalues and diagonal of inverse problems Pierre Carrier, Yousef Saad, James Freericks, Ehsan Khatami, Marcos Rigol, Tarek El- Ghazawi Description of the physical problems: 1. Numerical Linked-clusters (NLC) 2. (Real and imaginary time) Dynamical Mean-Field Theory (DMFT) Description of the numerical solvers: 3. NLC: eigenvalue problems with Lanczos 4. DMFT: diagonal of the inverse with Lanczos (consider also direct methods, probing,...) 5. Optimization and I/O of Lanczos basis vectors 1/16 Extreme Scale I/O and Data Analysis WorkshopMarch , Austin, Tx

2/16 Principles : Based on the linked-cluster basis of high-temperature expansions [Domb & Green], where analytical expansion in 1/T are replaced by an exact numerical calculation. C.Domb and M. S. Green Phase Transitions and Critical Phenomena (Academic Press, New York, 1974) J. Oitmaa, Ch. Hamer, andW. Zheng Series Expansion Methods for Strongly Interacting Lattice Models (Cambridge Univ. Press, Melbourne, 2006) M. Rigol, T. Bryant, and R. R. P. Singh Numerical Linked-Cluster Algorithms: I. Spin systems on square, triangular, and kagomé lattices Phys. Rev. E 75, (2007)Phys. Rev. E 75, (2007); arXiv: Numerical LinkNumerical Linked-Cluster Algorithms: II. t-J models on the square lattice Phys. Rev. E 75, (2007); arXiv: ; arXiv: Main references : Goal : Compute finite-temperature properties of generic quantum lattice systems at low temperatures. 1. Numerical-Linked clusters (NLC): physical problem

3/16 Goal : Compute finite-temperature properties of generic quantum lattice systems at low temperatures. Example : Phys. Rev. E 75, (2007) 3 sites4 sites....N sites 1. Numerical-Linked clusters (NLC): physical problem One example of sparse H : ,042 5,450 15,197 42, ,561 number of sites: number of clusters: Very large number of eigenvalue problems All eigenvalues of H are required H is sparse; pattern not fixed; dimension 2 N Very high-precision required ( ) c =

2. Dynamical Mean-Field Theory (DMFT): physical problem 4/16 James K. Freericks Transport in multilayered nanostructures, the dynamical mean-field theory approach (Imperial College Press, London, 2006) W. Metzner and D. Volhardt, Correlated Lattice Fermions in d=infinity dimensions Phys. Rev. Lett. 62, 324 (1988) G. Kotliar and D. Vollhardt Strongly Correlated Materials: Insights from Dynamical Mean-Field Theory Physics Today 57, No. 3 (March), 53 (2004); References therein. Main references : Goal : Compute fermion-boson many-body interactions for ultracold atoms in optical lattices. Imaginary time: Real time: Principles : Based on a mapping of the lattice problem onto an impurity problem that mimics the motion of electrons via their hopping from site to site, by solving the diagonal of the inverse of Dyson’s equations:

2. Dynamical Mean-Field Theory (DMFT): physical problem 5/16 Imaginary time: Real time: Off-diagonal elements are Hopping matrix t ab forms the off-diagonal elements and takes values 0 or 1, only G is complex symmetric non-hermitian Size of system is very large Several values of need to be evaluated G is sparse; pattern fixed moderate precision required (10 -8 ) Imaginary time problem is diagonally dominant

2. Dynamical Mean-Field Theory (DMFT): physical problem 6/16 DMFT loop (e.g., imaginary time) : Impurity solver: Continuation: Chemical potential: (for imaginary time only) Principles : Based on a mapping of the lattice problem onto an impurity problem that mimics the motion of electrons via their hopping from site to site, by solving the diagonal of the inverse of Dyson’s equations:

Lanczos approach using out-of-core memory for eigenvalues and diagonal of inverse problems Pierre Carrier, Yousef Saad, James Freericks, Ehsan Khatami, Marcos Rigol, Tarek El- Ghazawi Description of the physical problems: 1. Numerical Linked-clusters (NLC) 2. (Real and imaginary time) Dynamical Mean-Field Theory (DMFT) Description of the numerical solvers: 3. NLC: eigenvalue problems with Lanczos 4. DMFT: diagonal of the inverse with Lanczos 5. Optimization and I/O of Lanczos basis vectors 7/16

3. NLC: eigenvalue problems with Lanczos : numerical solvers Sparse matrix: Tridiagonal matrix: Much easier to diagonalize 8/16

3. NLC: eigenvalue problems with Lanczos : numerical solvers 9/16 Initialization Update Lanczos’ recurrence Simon’s and Kahan’s re-orthogonalization schemes

3. NLC: eigenvalue problems with Lanczos : numerical solvers 9/16 Initialization Update Lanczos’ recurrence Simon’s and Kahan’s re-orthogonalization schemes

4. DMFT: diagonal of the inverse with Lanczos 10/16

4. DMFT: diagonal of the inverse with Lanczos 11/16 After some algebra, one gets the diaginv algorithm... T m decomposition is

4. DMFT: diagonal of the inverse with Lanczos 12/16 Lanczos routine diaginv routine

13/16 Example of the diagonal of inverse Green’s function small matrix of dimension 441 X DMFT: diagonal of the inverse with Lanczos

Lanczos approach using out-of-core memory for eigenvalues and diagonal of inverse problems Pierre Carrier, Yousef Saad, James Freericks, Ehsan Khatami, Marcos Rigol, Tarek El- Ghazawi Description of the physical problems: 1. Numerical Linked-clusters (NLC) 2. (Real and imaginary time) Dynamical Mean-Field Theory (DMFT) Description of the numerical solvers: 3. NLC: eigenvalue problems with Lanczos 4. DMFT: diagonal of the inverse with Lanczos 5. Optimization and I/O of Lanczos basis vectors (beginning of project: Sept ’09) 14/16 All algorithms have been tested on the 2D problems and give accurate solutions, relatively fast

5. Optimization and I/O of Lanczos basis vectors 15/16 Impurity solver: Continuation: Chemical potential: (for imaginary time only) Lanczos routine diaginv routine swap?

16/16 I/O of Lanczos vectors Compression (wavelets) 5. Optimization and I/O of Lanczos basis vectors Load vectors by blocks from disk