Thermal tensor renormalization group simulations of square-lattice quantum spin models
Abstract
In this work, we benchmark the well-controlled and numerically accurate exponential thermal tensor renormalization group (XTRG) in the simulation of interacting spin models in two dimensions. Finite temperature introduces a finite thermal correlation length ξ, such that for system sizes L >> ξ finite-size calculations actually simulate the thermodynamic limit. Here in this paper, we focus on the square lattice Heisenberg antiferromagnet (SLH) and quantum Ising models (QIM) on open and cylindrical geometries up to width W = 10 . We explore various one-dimensional mapping paths in the matrix product operator (MPO) representation, whose performance is clearly shown to be geometry dependent. We benchmark against quantum Monte Carlo (QMC) data, yet also the series-expansion thermal tensor network results. Thermal properties including the internal energy, specific heat, and spin structure factors, etc. are computed with high precision, obtaining excellent agreement with QMC results. XTRG also allows us to reach remarkably low temperatures. For SLH, we obtain an energy per site u$$^*_g$$ ≃ -0.6694 (4) and a spontaneous magnetization m$$^*_S$$ ≃ 0.30 ( 1 ) already consistent with the ground-state properties, which is obtained from extrapolated low-T thermal data on W ≤ 8 cylinders and W ≤ 10 open strips, respectively. We extract an exponential divergence versus T of the structure factor S (M), as well as the correlation length ξ, at the ordering wave vector M = ($π,π$) , which represents the renormalized classical behavior and can be observed over a narrow but appreciable temperature window, by analyzing the finite-size data by XTRG simulations. For the QIM with a finite-temperature phase transition, we employ several thermal quantities, including the specific heat, Binder ratio, as well as the MPO entanglement to determine the critical temperature $$T_c$$.
- Authors:
-
- Beihang Univ., Beijing (China)
- Beihang Univ., Beijing (China); Ludwig-Maximilians-Univ., München (Germany)
- Ludwig-Maximilians-Univ., München (Germany)
- Brookhaven National Lab. (BNL), Upton, NY (United States); Ludwig-Maximilians-Univ., München (Germany)
- Publication Date:
- Research Org.:
- Brookhaven National Lab. (BNL), Upton, NY (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES)
- OSTI Identifier:
- 1543401
- Alternate Identifier(s):
- OSTI ID: 1546376
- Report Number(s):
- BNL-211858-2019-JAAM
Journal ID: ISSN 2469-9950; PRBMDO
- Grant/Contract Number:
- SC0012704
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review B
- Additional Journal Information:
- Journal Volume: 100; Journal Issue: 4; Journal ID: ISSN 2469-9950
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY
Citation Formats
Li, Han, Chen, Bin-Bin, Chen, Ziyu, von Delft, Jan, Weichselbaum, Andreas, and Li, Wei. Thermal tensor renormalization group simulations of square-lattice quantum spin models. United States: N. p., 2019.
Web. doi:10.1103/PhysRevB.100.045110.
Li, Han, Chen, Bin-Bin, Chen, Ziyu, von Delft, Jan, Weichselbaum, Andreas, & Li, Wei. Thermal tensor renormalization group simulations of square-lattice quantum spin models. United States. https://doi.org/10.1103/PhysRevB.100.045110
Li, Han, Chen, Bin-Bin, Chen, Ziyu, von Delft, Jan, Weichselbaum, Andreas, and Li, Wei. Tue .
"Thermal tensor renormalization group simulations of square-lattice quantum spin models". United States. https://doi.org/10.1103/PhysRevB.100.045110. https://www.osti.gov/servlets/purl/1543401.
@article{osti_1543401,
title = {Thermal tensor renormalization group simulations of square-lattice quantum spin models},
author = {Li, Han and Chen, Bin-Bin and Chen, Ziyu and von Delft, Jan and Weichselbaum, Andreas and Li, Wei},
abstractNote = {In this work, we benchmark the well-controlled and numerically accurate exponential thermal tensor renormalization group (XTRG) in the simulation of interacting spin models in two dimensions. Finite temperature introduces a finite thermal correlation length ξ, such that for system sizes L >> ξ finite-size calculations actually simulate the thermodynamic limit. Here in this paper, we focus on the square lattice Heisenberg antiferromagnet (SLH) and quantum Ising models (QIM) on open and cylindrical geometries up to width W = 10 . We explore various one-dimensional mapping paths in the matrix product operator (MPO) representation, whose performance is clearly shown to be geometry dependent. We benchmark against quantum Monte Carlo (QMC) data, yet also the series-expansion thermal tensor network results. Thermal properties including the internal energy, specific heat, and spin structure factors, etc. are computed with high precision, obtaining excellent agreement with QMC results. XTRG also allows us to reach remarkably low temperatures. For SLH, we obtain an energy per site u$^*_g$ ≃ -0.6694 (4) and a spontaneous magnetization m$^*_S$ ≃ 0.30 ( 1 ) already consistent with the ground-state properties, which is obtained from extrapolated low-T thermal data on W ≤ 8 cylinders and W ≤ 10 open strips, respectively. We extract an exponential divergence versus T of the structure factor S (M), as well as the correlation length ξ, at the ordering wave vector M = ($π,π$) , which represents the renormalized classical behavior and can be observed over a narrow but appreciable temperature window, by analyzing the finite-size data by XTRG simulations. For the QIM with a finite-temperature phase transition, we employ several thermal quantities, including the specific heat, Binder ratio, as well as the MPO entanglement to determine the critical temperature $T_c$.},
doi = {10.1103/PhysRevB.100.045110},
journal = {Physical Review B},
number = 4,
volume = 100,
place = {United States},
year = {2019},
month = {7}
}
Web of Science
Figures / Tables:

Works referenced in this record:
Two-dimensional quantum Heisenberg antiferromagnet at low temperatures
journal, February 1989
- Chakravarty, Sudip; Halperin, Bertrand I.; Nelson, David R.
- Physical Review B, Vol. 39, Issue 4
Spin correlations in the 2D Heisenberg antiferromagnet : Neutron scattering, Monte Carlo simulation, and theory
journal, February 1994
- Greven, M.; Birgeneau, R. J.; Endoh, Y.
- Physical Review Letters, Vol. 72, Issue 7
Spin Dependence of Correlations in Two-Dimensional Square-Lattice Quantum Heisenberg Antiferromagnets
journal, July 1995
- Elstner, N.; Sokol, A.; Singh, R. R. P.
- Physical Review Letters, Vol. 75, Issue 5
Complexity in Strongly Correlated Electronic Systems
journal, July 2005
- Dagotto, E.
- Science, Vol. 309, Issue 5732
: A spin- triangular-lattice Heisenberg antiferromagnet in the two-dimensional limit
journal, February 2017
- Rawl, R.; Ge, L.; Agrawal, H.
- Physical Review B, Vol. 95, Issue 6
Density matrix formulation for quantum renormalization groups
journal, November 1992
- White, Steven R.
- Physical Review Letters, Vol. 69, Issue 19
Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems
journal, March 2008
- Verstraete, F.; Murg, V.; Cirac, J. I.
- Advances in Physics, Vol. 57, Issue 2
Spin-Liquid Ground State of the S = 1/2 Kagome Heisenberg Antiferromagnet
journal, April 2011
- Yan, S.; Huse, D. A.; White, S. R.
- Science, Vol. 332, Issue 6034
Nature of the Spin-Liquid Ground State of the Heisenberg Model on the Kagome Lattice
journal, August 2012
- Depenbrock, Stefan; McCulloch, Ian P.; Schollwöck, Ulrich
- Physical Review Letters, Vol. 109, Issue 6
Neél Order in Square and Triangular Lattice Heisenberg Models
journal, September 2007
- White, Steven R.; Chernyshev, A. L.
- Physical Review Letters, Vol. 99, Issue 12
Spin liquid phase of the Heisenberg model on the triangular lattice
journal, July 2015
- Zhu, Zhenyue; White, Steven R.
- Physical Review B, Vol. 92, Issue 4
Competing spin-liquid states in the spin- Heisenberg model on the triangular lattice
journal, October 2015
- Hu, Wen-Jun; Gong, Shou-Shu; Zhu, Wei
- Physical Review B, Vol. 92, Issue 14
Topography of Spin Liquids on a Triangular Lattice
journal, May 2018
- Zhu, Zhenyue; Maksimov, P. A.; White, Steven R.
- Physical Review Letters, Vol. 120, Issue 20
The density matrix renormalization group for a quantum spin chain at non-zero temperature
journal, September 1996
- Bursill, R. J.; Xiang, T.; Gehring, G. A.
- Journal of Physics: Condensed Matter, Vol. 8, Issue 40
Transfer-matrix density-matrix renormalization-group theory for thermodynamics of one-dimensional quantum systems
journal, September 1997
- Wang, Xiaoqun; Xiang, Tao
- Physical Review B, Vol. 56, Issue 9
Thermodynamics of quantum Heisenberg spin chains
journal, October 1998
- Xiang, Tao
- Physical Review B, Vol. 58, Issue 14
Finite-temperature density matrix renormalization using an enlarged Hilbert space
journal, December 2005
- Feiguin, Adrian E.; White, Steven R.
- Physical Review B, Vol. 72, Issue 22
Minimally Entangled Typical Quantum States at Finite Temperature
journal, May 2009
- White, Steven R.
- Physical Review Letters, Vol. 102, Issue 19
Minimally entangled typical thermal state algorithms
journal, May 2010
- Stoudenmire, E. M.; White, Steven R.
- New Journal of Physics, Vol. 12, Issue 5
Linearized Tensor Renormalization Group Algorithm for the Calculation of Thermodynamic Properties of Quantum Lattice Models
journal, March 2011
- Li, Wei; Ran, Shi-Ju; Gong, Shou-Shu
- Physical Review Letters, Vol. 106, Issue 12
Bilayer linearized tensor renormalization group approach for thermal tensor networks
journal, April 2017
- Dong, Yong-Liang; Chen, Lei; Liu, Yun-Jing
- Physical Review B, Vol. 95, Issue 14
Optimized decimation of tensor networks with super-orthogonalization for two-dimensional quantum lattice models
journal, October 2012
- Ran, Shi-Ju; Li, Wei; Xi, Bin
- Physical Review B, Vol. 86, Issue 13
Projected entangled pair states at finite temperature: Imaginary time evolution with ancillas
journal, December 2012
- Czarnik, Piotr; Cincio, Lukasz; Dziarmaga, Jacek
- Physical Review B, Vol. 86, Issue 24
Variational approach to projected entangled pair states at finite temperature
journal, July 2015
- Czarnik, Piotr; Dziarmaga, Jacek
- Physical Review B, Vol. 92, Issue 3
Variational tensor network renormalization in imaginary time: Benchmark results in the Hubbard model at finite temperature
journal, December 2016
- Czarnik, Piotr; Rams, Marek M.; Dziarmaga, Jacek
- Physical Review B, Vol. 94, Issue 23
Overcoming the sign problem at finite temperature: Quantum tensor network for the orbital model on an infinite square lattice
journal, July 2017
- Czarnik, Piotr; Dziarmaga, Jacek; Oleś, Andrzej M.
- Physical Review B, Vol. 96, Issue 1
Time evolution of an infinite projected entangled pair state: An efficient algorithm
journal, January 2019
- Czarnik, Piotr; Dziarmaga, Jacek; Corboz, Philippe
- Physical Review B, Vol. 99, Issue 3
Finite correlation length scaling with infinite projected entangled pair states at finite temperature
journal, June 2019
- Czarnik, Piotr; Corboz, Philippe
- Physical Review B, Vol. 99, Issue 24
Series-expansion thermal tensor network approach for quantum lattice models
journal, April 2017
- Chen, Bin-Bin; Liu, Yun-Jing; Chen, Ziyu
- Physical Review B, Vol. 95, Issue 16
Exponential Thermal Tensor Network Approach for Quantum Lattice Models
journal, September 2018
- Chen, Bin-Bin; Chen, Lei; Chen, Ziyu
- Physical Review X, Vol. 8, Issue 3
Two-temperature scales in the triangular-lattice Heisenberg antiferromagnet
journal, April 2019
- Chen, Lei; Qu, Dai-Wei; Li, Han
- Physical Review B, Vol. 99, Issue 14
An Approximate Quantum Theory of the Antiferromagnetic Ground State
journal, June 1952
- Anderson, P. W.
- Physical Review, Vol. 86, Issue 5
Monte Carlo simulations of the spin-(1/2 Heisenberg antiferromagnet on a square lattice
journal, April 1988
- Reger, J. D.; Young, A. P.
- Physical Review B, Vol. 37, Issue 10
Simple Variational Wave Functions for Two-Dimensional Heisenberg Spin-½ Antiferromagnets
journal, June 1988
- Huse, David A.; Elser, Veit
- Physical Review Letters, Vol. 60, Issue 24
Some New Variational Resonating-Valence-Bond-Type Wave Functions for the Spin-½ Antiferromagnetic Heisenberg Model on a Square Lattice
journal, July 1988
- Liang, S.; Doucot, B.; Anderson, P. W.
- Physical Review Letters, Vol. 61, Issue 3
Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models
journal, November 1966
- Mermin, N. D.; Wagner, H.
- Physical Review Letters, Vol. 17, Issue 22
The spin-½ Heisenberg antiferromagnet on a square lattice and its application to the cuprous oxides
journal, January 1991
- Manousakis, Efstratios
- Reviews of Modern Physics, Vol. 63, Issue 1
Cluster Monte Carlo simulation of the transverse Ising model
journal, December 2002
- Blöte, Henk W. J.; Deng, Youjin
- Physical Review E, Vol. 66, Issue 6
Spectral functions in one-dimensional quantum systems at finite temperature using the density matrix renormalization group
journal, June 2009
- Barthel, Thomas; Schollwöck, Ulrich; White, Steven R.
- Physical Review B, Vol. 79, Issue 24
Mixed-State Dynamics in One-Dimensional Quantum Lattice Systems: A Time-Dependent Superoperator Renormalization Algorithm
journal, November 2004
- Zwolak, Michael; Vidal, Guifré
- Physical Review Letters, Vol. 93, Issue 20
Finite temperature properties of the spin-1/2 Heisenberg antiferromagnet on the triangular lattice
journal, September 1993
- Elstner, N.; Singh, R. R. P.; Young, A. P.
- Physical Review Letters, Vol. 71, Issue 10
The density-matrix renormalization group in the age of matrix product states
journal, January 2011
- Schollwöck, Ulrich
- Annals of Physics, Vol. 326, Issue 1
Entanglement scaling of operators: a conformal field theory approach, with a glimpse of simulability of long-time dynamics in 1 + 1d
journal, May 2017
- Dubail, J.
- Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 23
Operator space entanglement entropy in a transverse Ising chain
journal, September 2007
- Prosen, Tomaž; Pižorn, Iztok
- Physical Review A, Vol. 76, Issue 3
Non-abelian symmetries in tensor networks: A quantum symmetry space approach
journal, December 2012
- Weichselbaum, Andreas
- Annals of Physics, Vol. 327, Issue 12
Two-dimensional algorithm of the density-matrix renormalization group
journal, August 2001
- Xiang, Tao; Lou, Jizhong; Su, Zhaobin
- Physical Review B, Vol. 64, Issue 10
The ALPS project release 2.0: open source software for strongly correlated systems
journal, May 2011
- Bauer, B.; Carr, L. D.; Evertz, H. G.
- Journal of Statistical Mechanics: Theory and Experiment, Vol. 2011, Issue 05
Monte Carlo Study of Quantum spin Systems on the Square Lattice
journal, December 1988
- Okabe, Y.; Kikuchi, M.
- Le Journal de Physique Colloques, Vol. 49, Issue C8
Signature of a quantum dimensional transition in the spin- antiferromagnetic Heisenberg model on a square lattice and space reduction in the matrix product state
journal, April 2019
- Wang, Lihua; Kim, Kwang S.
- Physical Review B, Vol. 99, Issue 13
A cold-atom Fermi–Hubbard antiferromagnet
journal, May 2017
- Mazurenko, Anton; Chiu, Christie S.; Ji, Geoffrey
- Nature, Vol. 545, Issue 7655
Finite-size scaling of the ground-state parameters of the two-dimensional Heisenberg model
journal, November 1997
- Sandvik, Anders W.
- Physical Review B, Vol. 56, Issue 18
Low-temperature behavior of two-dimensional quantum antiferromagnets
journal, March 1988
- Chakravarty, Sudip; Halperin, Bertrand I.; Nelson, David R.
- Physical Review Letters, Vol. 60, Issue 11
Square-Lattice Heisenberg Antiferromagnet at Very Large Correlation Lengths
journal, February 1998
- Beard, B. B.; Birgeneau, R. J.; Greven, M.
- Physical Review Letters, Vol. 80, Issue 8
Low Temperature Behavior and Crossovers of the Square Lattice Quantum Heisenberg Antiferromagnet
journal, March 1998
- Kim, Jae-Kwon; Troyer, Matthias
- Physical Review Letters, Vol. 80, Issue 12
Spin‐1/2 Heisenberg antiferromagnet on the square and triangular lattices: A comparison of finite temperature properties
journal, May 1994
- Elstner, N.; Singh, R. R. P.; Young, A. P.
- Journal of Applied Physics, Vol. 75, Issue 10
Complexity of thermal states in quantum spin chains
journal, August 2008
- Žnidarič, Marko; Prosen, Tomaž; Pižorn, Iztok
- Physical Review A, Vol. 78, Issue 2
Anomalies in the entanglement properties of the square-lattice Heisenberg model
journal, October 2011
- Kallin, Ann B.; Hastings, Matthew B.; Melko, Roger G.
- Physical Review B, Vol. 84, Issue 16
Entanglement entropy of the two-dimensional Heisenberg antiferromagnet
journal, June 2011
- Song, H. Francis; Laflorencie, Nicolas; Rachel, Stephan
- Physical Review B, Vol. 83, Issue 22
Quantum Monte Carlo calculation of entanglement Rényi entropies for generic quantum systems
journal, December 2012
- Humeniuk, Stephan; Roscilde, Tommaso
- Physical Review B, Vol. 86, Issue 23
Detecting Goldstone modes with entanglement entropy
journal, September 2015
- Kulchytskyy, Bohdan; Herdman, C. M.; Inglis, Stephen
- Physical Review B, Vol. 92, Issue 11
Universal logarithmic corrections to entanglement entropies in two dimensions with spontaneously broken continuous symmetries
journal, April 2015
- Luitz, David J.; Plat, Xavier; Alet, Fabien
- Physical Review B, Vol. 91, Issue 15
Spin-wave approach for entanglement entropies of the Heisenberg antiferromagnet on the square lattice
journal, September 2015
- Laflorencie, Nicolas; Luitz, David J.; Alet, Fabien
- Physical Review B, Vol. 92, Issue 11
Simulation of two-dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction
journal, September 2009
- Orús, Román; Vidal, Guifré
- Physical Review B, Vol. 80, Issue 9
Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems
journal, May 2012
- Orús, Román
- Physical Review B, Vol. 85, Issue 20
Thermal Ising transitions in the vicinity of two-dimensional quantum critical points
journal, April 2016
- Hesselmann, S.; Wessel, S.
- Physical Review B, Vol. 93, Issue 15
Critical Properties from Monte Carlo Coarse Graining and Renormalization
journal, August 1981
- Binder, K.
- Physical Review Letters, Vol. 47, Issue 9
Finite size scaling analysis of ising model block distribution functions
journal, June 1981
- Binder, K.
- Zeitschrift f�r Physik B Condensed Matter, Vol. 43, Issue 2
Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models
journal, December 1966
- Mermin, N. D.; Wagner, H.
- Physical Review Letters, Vol. 17, Issue 26
Works referencing / citing this record:
Tensor network simulation of the Kitaev-Heisenberg model at finite temperature
journal, October 2019
- Czarnik, Piotr; Francuz, Anna; Dziarmaga, Jacek
- Physical Review B, Vol. 100, Issue 16
Thermodynamic properties of the Shastry-Sutherland model throughout the dimer-product phase
journal, October 2019
- Wietek, Alexander; Corboz, Philippe; Wessel, Stefan
- Physical Review Research, Vol. 1, Issue 3
Thermodynamic properties of the Shastry-Sutherland model throughout the dimer-product phase
text, January 2019
- Wietek, Alexander; Corboz, Philippe; Weßel, Stefan
- RWTH Aachen University
Figures / Tables found in this record: