We propose a series-expansion thermal tensor network (SETTN) approach for efficient simulations of quantum lattice models. This continuous-time SETTN method is based on the numerically exact Taylor series expansion of the equilibrium density operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ (with $H$ the total Hamiltonian and $\ensuremath{\beta}$ the imaginary time), and is thus Trotter-error free. We discover, through simulating XXZ spin chain and square-lattice quantum Ising models, that not only the Hamiltonian $H$ , but also its powers ${H}^{n}$ , can be efficiently expressed as matrix product operators, which enables us to calculate with high precision the equilibrium and dynamical properties of quantum lattice models at finite temperatures. Our SETTN method provides an alternative to conventional Trotter-Suzuki renormalization-group (RG) approaches, and achieves a very high standard of thermal RG simulations in terms of accuracy and flexibility.We propose a series-expansion thermal tensor network (SETTN) approach for efficient simulations of quantum lattice models. This continuous-time SETTN method is based on the numerically exact Taylor series expansion of the equilibrium density operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ (with $H$ the total Hamiltonian and $\ensuremath{\beta}$ the imaginary time), and is thus Trotter-error free. We discover, through simulating XXZ spin chain and square-lattice quantum Ising models, that not only the Hamiltonian $H$, but also its powers ${H}^{n}$, can be efficiently expressed as matrix product operators, which enables us to calculate with high precision the equilibrium and dynamical properties of quantum lattice models at finite temperatures. Our SETTN method provides an alternative to conventional Trotter-Suzuki renormalization-group (RG) approaches, and achieves a very high standard of thermal RG simulations in terms of accuracy and flexibility.

arXiv:1609.01263Physical review. B./Physical review. BApr 7, 201740 citations

Tensor network methods for quantum lattice systems

Ashley Milsted

Jan 1, 20160 citations

Continuous-Time Thermal Tensor Network Approach for Quantum Lattice Models

Bin-Bin Chen, Yun-Jing Liu, Ziyu Chen +1 more

In this Letter we propose a continuous-time thermal tensor network (TTN) approach for efficient simulations of quantum lattice models. The continuous-time TTN method is based on numerically exact Taylor series expansions of equilibrium density operators $e^{-\beta H}$ (with $H$ the total Hamiltonian and $\beta$ the imaginary time), as well as unitary time-evolution operators $e^{-iHt}$ , and is thus Trotter-error free. We discover, through simulating a typical XXZ spin chain, that not only the Hamiltonian $H$ , but also its powers $H^n$ , can be efficiently expressed as matrix product operators, which enables us to calculate with high precision the equilibrium and dynamical properties of quantum lattice models at finite temperatures. Our series-expansion TTN method provides an alternative to conventional Trotter-Suzuki renormalization group (RG) approaches, and achieves an unprecedented standard of thermal RG simulations in terms of accuracy and flexibility.In this Letter we propose a continuous-time thermal tensor network (TTN) approach for efficient simulations of quantum lattice models. The continuous-time TTN method is based on numerically exact Taylor series expansions of equilibrium density operators $e^{-\beta H}$ (with $H$ the total Hamiltonian and $\beta$ the imaginary time), as well as unitary time-evolution operators $e^{-iHt}$, and is thus Trotter-error free. We discover, through simulating a typical XXZ spin chain, that not only the Hamiltonian $H$, but also its powers $H^n$, can be efficiently expressed as matrix product operators, which enables us to calculate with high precision the equilibrium and dynamical properties of quantum lattice models at finite temperatures. Our series-expansion TTN method provides an alternative to conventional Trotter-Suzuki renormalization group (RG) approaches, and achieves an unprecedented standard of thermal RG simulations in terms of accuracy and flexibility.

Sep 5, 20160 citations

Variational tensor network renormalization in imaginary time: Benchmark results in the Hubbard model at finite temperature

Piotr Czarnik, Marek M. Rams, Jacek Dziarmaga

A Gibbs operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ for a two-dimensional (2D) lattice system with a Hamiltonian $H$ can be represented by a 3D tensor network, with the third dimension being the imaginary time (inverse temperature) $\ensuremath{\beta}$. Coarse graining the network along $\ensuremath{\beta}$ results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension. The coarse graining is performed by a tree tensor network of isometries. They are optimized variationally to maximize the accuracy of the PEPO as a representation of the 2D thermal state ${e}^{\ensuremath{-}\ensuremath{\beta}H}$. The algorithm is applied to the two-dimensional Hubbard model on an infinite square lattice. Benchmark results at finite temperature are obtained that are consistent with the best cluster dynamical mean-field theory and power-series expansion in the regime of parameters where they yield mutually consistent results.A Gibbs operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ for a two-dimensional (2D) lattice system with a Hamiltonian $H$ can be represented by a 3D tensor network, with the third dimension being the imaginary time (inverse temperature) $\ensuremath{\beta}$. Coarse graining the network along $\ensuremath{\beta}$ results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension. The coarse graining is performed by a tree tensor network of isometries. They are optimized variationally to maximize the accuracy of the PEPO as a representation of the 2D thermal state ${e}^{\ensuremath{-}\ensuremath{\beta}H}$. The algorithm is applied to the two-dimensional Hubbard model on an infinite square lattice. Benchmark results at finite temperature are obtained that are consistent with the best cluster dynamical mean-field theory and power-series expansion in the regime of parameters where they yield mutually consistent results.

arXiv:1607.04016Physical review. B./Physical review. BDec 16, 201669 citations

NOVEL TENSOR NETWORK RENORMALIZATION GROUP APPROACHES FOR QUANTUM LATTICE MODELS

Gang Su

WORLD SCIENTIFIC eBooksAug 14, 20130 citations

H

the total Hamiltonian and $\ensuremath{\beta}$ the imaginary time), and is thus Trotter-error free. We discover, through simulating XXZ spin chain and square-lattice quantum Ising models, that not only the Hamiltonian

H

, but also its powers

{H}^{n}

, can be efficiently expressed as matrix product operators, which enables us to calculate with high precision the equilibrium and dynamical properties of quantum lattice models at finite temperatures. Our SETTN method provides an alternative to conventional Trotter-Suzuki renormalization-group (RG) approaches, and achieves a very high standard of thermal RG simulations in terms of accuracy and flexibility.

e^{-\beta H}

(with

H

the total Hamiltonian and

\beta

the imaginary time), as well as unitary time-evolution operators

e^{-iHt}

, and is thus Trotter-error free. We discover, through simulating a typical XXZ spin chain, that not only the Hamiltonian

H

, but also its powers

H^n

, can be efficiently expressed as matrix product operators, which enables us to calculate with high precision the equilibrium and dynamical properties of quantum lattice models at finite temperatures. Our series-expansion TTN method provides an alternative to conventional Trotter-Suzuki renormalization group (RG) approaches, and achieves an unprecedented standard of thermal RG simulations in terms of accuracy and flexibility.

A Gibbs operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ for a two-dimensional (2D) lattice system with a Hamiltonian

H

can be represented by a 3D tensor network, with the third dimension being the imaginary time (inverse temperature) $\ensuremath{\beta}$. Coarse graining the network along $\ensuremath{\beta}$ results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension. The coarse graining is performed by a tree tensor network of isometries. They are optimized variationally to maximize the accuracy of the PEPO as a representation of the 2D thermal state ${e}^{\ensuremath{-}\ensuremath{\beta}H}$. The algorithm is applied to the two-dimensional Hubbard model on an infinite square lattice. Benchmark results at finite temperature are obtained that are consistent with the best cluster dynamical mean-field theory and power-series expansion in the regime of parameters where they yield mutually consistent results.