Exponential Thermal Tensor Network Approach for Quantum Lattice Models

We speed up thermal simulations of quantum many-body systems in both one- (1D) and two-dimensional (2D) models in an exponential way by iteratively projecting the thermal density matrix ρ[over ^]=e^{-βH[over ^]} onto itself. We refer to this scheme of doubling β in each step of the imaginary time ev...

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Autores principales:	Bin-Bin Chen, Lei Chen, Ziyu Chen, Wei Li, Andreas Weichselbaum
Formato:	article
Lenguaje:	EN
Publicado:	American Physical Society 2018
Materias:	Physics QC1-999
Acceso en línea:	https://doaj.org/article/5873e6f57eef4ae7a853502b80da7538
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spelling	oai:doaj.org-article:5873e6f57eef4ae7a853502b80da75382021-12-02T16:55:37ZExponential Thermal Tensor Network Approach for Quantum Lattice Models10.1103/PhysRevX.8.0310822160-3308https://doaj.org/article/5873e6f57eef4ae7a853502b80da75382018-09-01T00:00:00Zhttp://doi.org/10.1103/PhysRevX.8.031082http://doi.org/10.1103/PhysRevX.8.031082https://doaj.org/toc/2160-3308We speed up thermal simulations of quantum many-body systems in both one- (1D) and two-dimensional (2D) models in an exponential way by iteratively projecting the thermal density matrix ρ[over ^]=e^{-βH[over ^]} onto itself. We refer to this scheme of doubling β in each step of the imaginary time evolution as the exponential tensor renormalization group (XTRG). This approach is in stark contrast to conventional Trotter-Suzuki-type methods which evolve ρ[over ^] on a linear quasicontinuous grid in inverse temperature β≡1/T. As an aside, the large steps in XTRG allow one to swiftly jump across finite-temperature phase transitions, i.e., without the need to resolve each singularly expensive phase-transition point right away, e.g., when interested in low-energy behavior. A fine temperature resolution can be obtained, nevertheless, by using interleaved temperature grids. In general, XTRG can reach low temperatures exponentially fast and, thus, not only saves computational time but also merits better accuracy due to significantly fewer truncation steps. For similar reasons, we also find that the series expansion thermal tensor network approach benefits in both efficiency and precision, from the logarithmic temperature scale setup. We work in an (effective) 1D setting exploiting matrix product operators (MPOs), which allows us to fully and uniquely implement non-Abelian and Abelian symmetries to greatly enhance numerical performance. We use our XTRG machinery to explore the thermal properties of Heisenberg models on 1D chains and 2D square and triangular lattices down to low temperatures approaching ground-state properties. The entanglement properties, as well as the renormalization-group flow of entanglement spectra in MPOs, are discussed, where logarithmic entropies (approximately lnβ) are shown in both spin chains and square-lattice models with gapless towers of states. We also reveal that XTRG can be employed to accurately simulate the Heisenberg XXZ model on the square lattice which undergoes a thermal phase transition. We determine its critical temperature based on thermal physical observables, as well as entanglement measures. Overall, we demonstrate that XTRG provides an elegant, versatile, and highly competitive approach to explore thermal properties, including finite-temperature thermal phase transitions as well as the different ordering tendencies at various temperature scales for frustrated systems.Bin-Bin ChenLei ChenZiyu ChenWei LiAndreas WeichselbaumAmerican Physical SocietyarticlePhysicsQC1-999ENPhysical Review X, Vol 8, Iss 3, p 031082 (2018)
institution	DOAJ
collection	DOAJ
language	EN
topic	Physics QC1-999
spellingShingle	Physics QC1-999 Bin-Bin Chen Lei Chen Ziyu Chen Wei Li Andreas Weichselbaum Exponential Thermal Tensor Network Approach for Quantum Lattice Models
description	We speed up thermal simulations of quantum many-body systems in both one- (1D) and two-dimensional (2D) models in an exponential way by iteratively projecting the thermal density matrix ρ[over ^]=e^{-βH[over ^]} onto itself. We refer to this scheme of doubling β in each step of the imaginary time evolution as the exponential tensor renormalization group (XTRG). This approach is in stark contrast to conventional Trotter-Suzuki-type methods which evolve ρ[over ^] on a linear quasicontinuous grid in inverse temperature β≡1/T. As an aside, the large steps in XTRG allow one to swiftly jump across finite-temperature phase transitions, i.e., without the need to resolve each singularly expensive phase-transition point right away, e.g., when interested in low-energy behavior. A fine temperature resolution can be obtained, nevertheless, by using interleaved temperature grids. In general, XTRG can reach low temperatures exponentially fast and, thus, not only saves computational time but also merits better accuracy due to significantly fewer truncation steps. For similar reasons, we also find that the series expansion thermal tensor network approach benefits in both efficiency and precision, from the logarithmic temperature scale setup. We work in an (effective) 1D setting exploiting matrix product operators (MPOs), which allows us to fully and uniquely implement non-Abelian and Abelian symmetries to greatly enhance numerical performance. We use our XTRG machinery to explore the thermal properties of Heisenberg models on 1D chains and 2D square and triangular lattices down to low temperatures approaching ground-state properties. The entanglement properties, as well as the renormalization-group flow of entanglement spectra in MPOs, are discussed, where logarithmic entropies (approximately lnβ) are shown in both spin chains and square-lattice models with gapless towers of states. We also reveal that XTRG can be employed to accurately simulate the Heisenberg XXZ model on the square lattice which undergoes a thermal phase transition. We determine its critical temperature based on thermal physical observables, as well as entanglement measures. Overall, we demonstrate that XTRG provides an elegant, versatile, and highly competitive approach to explore thermal properties, including finite-temperature thermal phase transitions as well as the different ordering tendencies at various temperature scales for frustrated systems.
format	article
author	Bin-Bin Chen Lei Chen Ziyu Chen Wei Li Andreas Weichselbaum
author_facet	Bin-Bin Chen Lei Chen Ziyu Chen Wei Li Andreas Weichselbaum
author_sort	Bin-Bin Chen
title	Exponential Thermal Tensor Network Approach for Quantum Lattice Models
title_short	Exponential Thermal Tensor Network Approach for Quantum Lattice Models
title_full	Exponential Thermal Tensor Network Approach for Quantum Lattice Models
title_fullStr	Exponential Thermal Tensor Network Approach for Quantum Lattice Models
title_full_unstemmed	Exponential Thermal Tensor Network Approach for Quantum Lattice Models
title_sort	exponential thermal tensor network approach for quantum lattice models
publisher	American Physical Society
publishDate	2018
url	https://doaj.org/article/5873e6f57eef4ae7a853502b80da7538
work_keys_str_mv	AT binbinchen exponentialthermaltensornetworkapproachforquantumlatticemodels AT leichen exponentialthermaltensornetworkapproachforquantumlatticemodels AT ziyuchen exponentialthermaltensornetworkapproachforquantumlatticemodels AT weili exponentialthermaltensornetworkapproachforquantumlatticemodels AT andreasweichselbaum exponentialthermaltensornetworkapproachforquantumlatticemodels
_version_	1718382801452007424

Exponential Thermal Tensor Network Approach for Quantum Lattice Models

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