Due to the unfavorable scaling of tensor-network methods with the refinement parameter $M$ , new approaches are necessary to improve the efficiency of numerical simulations based on such states, in particular for gapless, strongly entangled systems. In one-dimensional density matrix renormalization group methods, the use of Abelian symmetries has led to large computational gain. In higher-dimensional tensor networks, this is associated with significant technical efforts and additional approximations. We explain a formalism to implement such symmetries in two-dimensional tensor-network states and present benchmark results that confirm the validity of these approximations in the context of projected entangled-pair state algorithms.Due to the unfavorable scaling of tensor-network methods with the refinement parameter $M$, new approaches are necessary to improve the efficiency of numerical simulations based on such states, in particular for gapless, strongly entangled systems. In one-dimensional density matrix renormalization group methods, the use of Abelian symmetries has led to large computational gain. In higher-dimensional tensor networks, this is associated with significant technical efforts and additional approximations. We explain a formalism to implement such symmetries in two-dimensional tensor-network states and present benchmark results that confirm the validity of these approximations in the context of projected entangled-pair state algorithms.

arXiv:1010.3595Physical Review BMar 18, 2011126 citations

Tensor Network Methods

Simone Montangero

Springer eBooksJan 1, 20180 citations

Exploiting the Hermitian symmetry in tensor network algorithms

Oscar van Alphen, Stijn V. Kleijweg, Juraj Hašík +1 more

Exploiting symmetries in tensor network algorithms plays a key role for reducing the computational and memory costs. Here we explain how to incorporate the Hermitian symmetry in double-layer tensor networks, which naturally arise in methods based on projected entangled-pair states. For real-valued tensors the Hermitian symmetry defines a ${\mathbb{Z}}_{2}$ symmetry on the combined bra and ket auxiliary level of the tensors. By implementing this symmetry, a speedup of the computation time by up to a factor 4 can be achieved, while expectation values of observables and reduced density matrices remain Hermitian by construction. Benchmark results based on the corner transfer matrix renormalization group and higher-order tensor renormalization group are presented. We also discuss how to implement the Hermitian symmetry in the complex case, where a similar speedup can be achieved.Exploiting symmetries in tensor network algorithms plays a key role for reducing the computational and memory costs. Here we explain how to incorporate the Hermitian symmetry in double-layer tensor networks, which naturally arise in methods based on projected entangled-pair states. For real-valued tensors the Hermitian symmetry defines a ${\mathbb{Z}}_{2}$ symmetry on the combined bra and ket auxiliary level of the tensors. By implementing this symmetry, a speedup of the computation time by up to a factor 4 can be achieved, while expectation values of observables and reduced density matrices remain Hermitian by construction. Benchmark results based on the corner transfer matrix renormalization group and higher-order tensor renormalization group are presented. We also discuss how to implement the Hermitian symmetry in the complex case, where a similar speedup can be achieved.

arXiv:2410.11596Physical review. B./Physical review. BJan 3, 20253 citations

Tensor Train Decomposition

Grey Ballard, Tamara G. Kolda

Cambridge University Press eBooksJun 10, 20250 citations

Benchmarking global SU(2) symmetry in two-dimensional tensor network algorithms

Philipp Schmoll, Román Orús

We implement and benchmark tensor network algorithms with $SU(2)$ symmetry for systems in two spatial dimensions and in the thermodynamic limit. Specifically, we implement $SU(2)$ -invariant versions of the infinite projected entangled pair states and infinite projected entangled simplex states methods. Our implementation of $SU(2)$ symmetry follows the formalism based on fusion trees from Schmoll et al. [Ann. Phys. 419, 168232 (2020)]. In order to assess the utility of implementing $SU(2)$ symmetry, the algorithms are benchmarked for three models with different local spin: the spin-1 bilinear-biquadratic model on the square lattice, and the kagome Heisenberg antiferromagnets (KHAFs) for spin- $1/2$ and spin-2. We observe that the implementation of $SU(2)$ symmetry provides better energies in general than nonsymmetric simulations, with smooth scalings with respect to the number of parameters in the ansatz, and with the actual improvement depending on the specifics of the model. In particular, for the spin-2 KHAF model, our $SU(2)$ simulations are compatible with a quantum spin-liquid ground state.We implement and benchmark tensor network algorithms with $SU(2)$ symmetry for systems in two spatial dimensions and in the thermodynamic limit. Specifically, we implement $SU(2)$-invariant versions of the infinite projected entangled pair states and infinite projected entangled simplex states methods. Our implementation of $SU(2)$ symmetry follows the formalism based on fusion trees from Schmoll et al. [Ann. Phys. 419, 168232 (2020)]. In order to assess the utility of implementing $SU(2)$ symmetry, the algorithms are benchmarked for three models with different local spin: the spin-1 bilinear-biquadratic model on the square lattice, and the kagome Heisenberg antiferromagnets (KHAFs) for spin-$1/2$ and spin-2. We observe that the implementation of $SU(2)$ symmetry provides better energies in general than nonsymmetric simulations, with smooth scalings with respect to the number of parameters in the ansatz, and with the actual improvement depending on the specifics of the model. In particular, for the spin-2 KHAF model, our $SU(2)$ simulations are compatible with a quantum spin-liquid ground state.

arXiv:2005.02748Physical review. B./Physical review. BDec 1, 202011 citations

Due to the unfavorable scaling of tensor-network methods with the refinement parameter

M

, new approaches are necessary to improve the efficiency of numerical simulations based on such states, in particular for gapless, strongly entangled systems. In one-dimensional density matrix renormalization group methods, the use of Abelian symmetries has led to large computational gain. In higher-dimensional tensor networks, this is associated with significant technical efforts and additional approximations. We explain a formalism to implement such symmetries in two-dimensional tensor-network states and present benchmark results that confirm the validity of these approximations in the context of projected entangled-pair state algorithms.

{\mathbb{Z}}_{2}

symmetry on the combined bra and ket auxiliary level of the tensors. By implementing this symmetry, a speedup of the computation time by up to a factor 4 can be achieved, while expectation values of observables and reduced density matrices remain Hermitian by construction. Benchmark results based on the corner transfer matrix renormalization group and higher-order tensor renormalization group are presented. We also discuss how to implement the Hermitian symmetry in the complex case, where a similar speedup can be achieved.

We implement and benchmark tensor network algorithms with

SU(2)

symmetry for systems in two spatial dimensions and in the thermodynamic limit. Specifically, we implement

SU(2)

-invariant versions of the infinite projected entangled pair states and infinite projected entangled simplex states methods. Our implementation of

SU(2)

symmetry follows the formalism based on fusion trees from Schmoll et al. [Ann. Phys. 419, 168232 (2020)]. In order to assess the utility of implementing

SU(2)

symmetry, the algorithms are benchmarked for three models with different local spin: the spin-1 bilinear-biquadratic model on the square lattice, and the kagome Heisenberg antiferromagnets (KHAFs) for spin-

1/2

and spin-2. We observe that the implementation of

SU(2)

symmetry provides better energies in general than nonsymmetric simulations, with smooth scalings with respect to the number of parameters in the ansatz, and with the actual improvement depending on the specifics of the model. In particular, for the spin-2 KHAF model, our

SU(2)

simulations are compatible with a quantum spin-liquid ground state.