VUMPS

Detailed Description

Implementation of the Variational Uniform Matrix Product States for Abelian and non Abelian symmetries. All equation references are according to the follwing paper: arXiv:1701.07035 "Variational optimization algorithms for uniform matrix product states" (V. Zauner-Stauber, L. Vanderstraeten, M.T. Fishman, F. Verstraete, J. Haegeman, 2017). Much of the earlier code can be actually reused, the only major additions are:

• The calculation of the environment involves infinite contractions that are carried out by solving linear systems eq. 14 or eq. C25ab. This is handled in VUMPS/VumpsTransferMatrix.h.
• The solution of the linear systems requires an GMRES solver. A simple version GMResSolver is implemented in the ALGS git.
• It is possible to run the algorithm with a 2-site Hamiltonian (without an MPO), see algorithm 2. This involves the local updates eq. 11 and eq. 16 which are handled in VUMPS/VumpsPivotStuff.h. Note that this version is only implemented for Abelian symmetries.
• The algorithm requires the calculation of and from and (in standard DMRG, is decomposed into or ; and is multiplied to the next site). This requires the modified sweeping algorithms Umps::svdDecompose (eq. 19, 20) and Umps::polarDecompose (eq. 21ab, 22).
• For the overlap between two UMPS, one needs to calculate the dominant eigenvalue of the transfer matrix eq. A7. This is achieved with the ArnoldiSolver from the ALGS git.
• Static Structure Factor for unit cells

Otherwise:

• The local update with an MPO involve the contractions eq. C28 and eq. C29. These are easily handled with the already existing pivot classes (1-site and 0-site).

Arbitrary sizes of unit cells are implemented. (simply create an MPO of this size). Always remember to set OPEN_BC=false for VUMPS.

Classes
class	Umps< Symmetry, Scalar >
struct	MpoTransferMatrix< Symmetry, Scalar >
struct	MpoTransferVector< Symmetry, Scalar_ >
struct	PivumpsMatrix1< Symmetry, Scalar, MpoScalar >
struct	PivumpsMatrix0< Symmetry, Scalar, MpoScalar >
class	VumpsSolver< Symmetry, MpHamiltonian, Scalar >
struct	TransferMatrix< Symmetry, Scalar >

Functions
template<typename Symmetry , typename Scalar1 , typename Scalar2 >
void	HxV (const MpoTransferMatrix< Symmetry, Scalar1 > &H, const MpoTransferVector< Symmetry, Scalar2 > &Vin, MpoTransferVector< Symmetry, Scalar2 > &Vout)

Function Documentation

HxV()

template<typename Symmetry , typename Scalar1 , typename Scalar2 >

void HxV	(	const MpoTransferMatrix< Symmetry, Scalar1 > &	H,
		const MpoTransferVector< Symmetry, Scalar2 > &	Vin,
		MpoTransferVector< Symmetry, Scalar2 > &	Vout
	)

Matrix-vector multiplication in eq. (25ab)

Definition at line 99 of file VumpsMpoTransferMatrix.h.