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
Extreme Scale I/O and Data Analysis Workshop March 22-24 2010, Austin, Tx
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1. Numerical-Linked clusters (NLC): physical problem
Goal: Compute finite-temperature properties of generic quantum lattice systems at low temperatures.
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.
Main references:
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, 061118 (2007); arXiv:0706.3254
Numerical Linked-Cluster Algorithms: II. t-J models on the square lattice
Phys. Rev. E 75, 061119 (2007); arXiv:0706.3255
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1. Numerical-Linked clusters (NLC): physical problem
Goal: Compute finite-temperature properties of generic quantum lattice systems at low temperatures.
3 sites
Example:
c=1 2
3
4 sites
4
5
6
7
....N sites
8
9
number of sites: 1 2 3 4 5 6 7 8
9 10 11
12
13
14
15
16
number of clusters: 1 1 1 3 4 10 19 51 112 300 746 2,042 5,450 15,197 42,192 119,561
One example of sparse H:
•Very large number of eigenvalue problems
•All eigenvalues of H are required
•H is sparse; pattern not fixed; dimension 2N
•Very high-precision required (10-12)
Phys. Rev. E 75, 061118 (2007)
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2. Dynamical Mean-Field Theory (DMFT): physical problem
Goal: Compute fermion-boson many-body interactions for ultracold atoms in optical lattices.
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:
Imaginary time:
Real time:
Main references:
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
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2. Dynamical Mean-Field Theory (DMFT): physical problem
Off-diagonal elements are
Imaginary time:
Real time:
•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
Hopping matrix tab forms the
off-diagonal elements
and takes values 0 or 1, only
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2. Dynamical Mean-Field Theory (DMFT): physical problem
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:
DMFT loop (e.g., imaginary time):
Impurity solver:
Chemical potential:
(for imaginary time only)
Continuation:
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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
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3. NLC: eigenvalue problems with Lanczos: numerical solvers
Sparse matrix:
Tridiagonal matrix:
Much easier to diagonalize
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3. NLC: eigenvalue problems with Lanczos: numerical solvers
Initialization
Lanczos’ recurrence
Simon’s and Kahan’s
re-orthogonalization schemes
Update
9/16
3. NLC: eigenvalue problems with Lanczos: numerical solvers
Initialization
Lanczos’ recurrence
Simon’s and Kahan’s
re-orthogonalization schemes
Update
9/16
4. DMFT: diagonal of the inverse with Lanczos
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4. DMFT: diagonal of the inverse with Lanczos
Tm decomposition is
After some algebra,
http://www.msi.umn.edu/~carrierp/images/DIAGINV_Lanczos.pdf
one gets the diaginv algorithm...
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4. DMFT: diagonal of the inverse with Lanczos
Lanczos
routine
diaginv
routine
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4. DMFT: diagonal of the inverse with Lanczos
Example of the diagonal of inverse Green’s function
small matrix of dimension 441 X 441
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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
All algorithms have been tested on the 2D problems and give accurate solutions, relatively fast
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
5. Optimization and I/O of Lanczos basis vectors
Lanczos routine
swap?
diaginv routine
Impurity solver:
Chemical potential:
(for imaginary time only)
Continuation:
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5. Optimization and I/O of Lanczos basis vectors
I/O of Lanczos vectors
Compression (wavelets)
Load vectors by blocks
from disk
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