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Title: Exponential Thermal Tensor Network Approach for Quantum Lattice Models

Abstract

Here, 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^ = e –βH^ 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^ 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 usmore » 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.« less

Authors:
 [1];  [1];  [1];  [1]; ORCiD logo [2]
  1. Beihang Univ., Beijing (China)
  2. Brookhaven National Lab. (BNL), Upton, NY (United States); Ludwig-Maximilians-Univ., Munich (Germany)
Publication Date:
Research Org.:
Brookhaven National Laboratory (BNL), Upton, NY (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)
OSTI Identifier:
1474015
Alternate Identifier(s):
OSTI ID: 1480975
Report Number(s):
BNL-209353-2018-JAAM
Journal ID: ISSN 2160-3308; PRXHAE
Grant/Contract Number:  
SC0012704
Resource Type:
Published Article
Journal Name:
Physical Review. X
Additional Journal Information:
Journal Volume: 8; Journal Issue: 3; Journal ID: ISSN 2160-3308
Publisher:
American Physical Society
Country of Publication:
United States
Language:
English
Subject:
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

Citation Formats

Chen, Bin -Bin, Chen, Lei, Chen, Ziyu, Li, Wei, and Weichselbaum, Andreas. Exponential Thermal Tensor Network Approach for Quantum Lattice Models. United States: N. p., 2018. Web. doi:10.1103/PhysRevX.8.031082.
Chen, Bin -Bin, Chen, Lei, Chen, Ziyu, Li, Wei, & Weichselbaum, Andreas. Exponential Thermal Tensor Network Approach for Quantum Lattice Models. United States. doi:10.1103/PhysRevX.8.031082.
Chen, Bin -Bin, Chen, Lei, Chen, Ziyu, Li, Wei, and Weichselbaum, Andreas. Wed . "Exponential Thermal Tensor Network Approach for Quantum Lattice Models". United States. doi:10.1103/PhysRevX.8.031082.
@article{osti_1474015,
title = {Exponential Thermal Tensor Network Approach for Quantum Lattice Models},
author = {Chen, Bin -Bin and Chen, Lei and Chen, Ziyu and Li, Wei and Weichselbaum, Andreas},
abstractNote = {Here, 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^ = e–βH^ 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^ 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.},
doi = {10.1103/PhysRevX.8.031082},
journal = {Physical Review. X},
number = 3,
volume = 8,
place = {United States},
year = {2018},
month = {9}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI: 10.1103/PhysRevX.8.031082

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