Precise Extrapolation of the Correlation Function Asymptotics in Uniform Tensor Network States with Application to the Bose-Hubbard and XXZ Models
Abstract
We analyze the problem of extracting the correlation length from infinite matrix product states (MPS) and corner transfer matrix (CTM) simulations. When the correlation length is calculated directly from the transfer matrix, it is typically significantly underestimated for finite bond dimensions used in numerical simulation. This is true even when one considers ground states at a distance from the critical point. In this article we introduce extrapolation procedure to overcome this problem. To that end we quantify how much the dominant part of the MPS (as well as CTM) transfer matrix spectrum deviates from being continuous. The latter is necessary to capture the exact asymptotics of the correlation function where the exponential decay is typically modified by an additional algebraic term. By extrapolating such a refinement parameter to zero, we show that we are able to recover the exact value of the correlation length with high accuracy. In a generic setting, our method reduces the error by a factor of ~100 as compared to the results obtained without extrapolation and a factor of ~10 as compared to simple extrapolation schemes employing bond dimension. We test our approach in a number of solvable models both in 1D and 2D. Subsequently, wemore »
- Authors:
- Publication Date:
- Research Org.:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program
- OSTI Identifier:
- 1483456
- Alternate Identifier(s):
- OSTI ID: 1489951
- Report Number(s):
- LA-UR-18-20465
Journal ID: ISSN 2160-3308; PRXHAE; 041033
- Grant/Contract Number:
- 89233218CNA000001
- Resource Type:
- Published Article
- Journal Name:
- Physical Review. X
- Additional Journal Information:
- Journal Name: Physical Review. X Journal Volume: 8 Journal Issue: 4; Journal ID: ISSN 2160-3308
- Publisher:
- American Physical Society
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 36 MATERIALS SCIENCE; 97 MATHEMATICS AND COMPUTING; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Mathematics
Citation Formats
Rams, Marek M., Czarnik, Piotr, and Cincio, Lukasz. Precise Extrapolation of the Correlation Function Asymptotics in Uniform Tensor Network States with Application to the Bose-Hubbard and XXZ Models. United States: N. p., 2018.
Web. doi:10.1103/PhysRevX.8.041033.
Rams, Marek M., Czarnik, Piotr, & Cincio, Lukasz. Precise Extrapolation of the Correlation Function Asymptotics in Uniform Tensor Network States with Application to the Bose-Hubbard and XXZ Models. United States. https://doi.org/10.1103/PhysRevX.8.041033
Rams, Marek M., Czarnik, Piotr, and Cincio, Lukasz. Tue .
"Precise Extrapolation of the Correlation Function Asymptotics in Uniform Tensor Network States with Application to the Bose-Hubbard and XXZ Models". United States. https://doi.org/10.1103/PhysRevX.8.041033.
@article{osti_1483456,
title = {Precise Extrapolation of the Correlation Function Asymptotics in Uniform Tensor Network States with Application to the Bose-Hubbard and XXZ Models},
author = {Rams, Marek M. and Czarnik, Piotr and Cincio, Lukasz},
abstractNote = {We analyze the problem of extracting the correlation length from infinite matrix product states (MPS) and corner transfer matrix (CTM) simulations. When the correlation length is calculated directly from the transfer matrix, it is typically significantly underestimated for finite bond dimensions used in numerical simulation. This is true even when one considers ground states at a distance from the critical point. In this article we introduce extrapolation procedure to overcome this problem. To that end we quantify how much the dominant part of the MPS (as well as CTM) transfer matrix spectrum deviates from being continuous. The latter is necessary to capture the exact asymptotics of the correlation function where the exponential decay is typically modified by an additional algebraic term. By extrapolating such a refinement parameter to zero, we show that we are able to recover the exact value of the correlation length with high accuracy. In a generic setting, our method reduces the error by a factor of ~100 as compared to the results obtained without extrapolation and a factor of ~10 as compared to simple extrapolation schemes employing bond dimension. We test our approach in a number of solvable models both in 1D and 2D. Subsequently, we apply it to one-dimensional XXZ spin-3/2 and the Bose-Hubbard models in a massive regime in the vicinity of the Berezinskii-Kosterlitz-Thouless critical point. We then fit the scaling form of the correlation length and extract the position of the critical point and obtain results comparable or better than those of other state-of-the-art numerical methods. Finally, we show how the algebraic part of the correlation function asymptotics can be directly recovered from the scaling of the dominant form factor within our approach. Our method provides the means for detailed studies of phase diagrams of quantum models in 1D and, through the finite correlation length scaling of projected entangled pair states, also in 2D.},
doi = {10.1103/PhysRevX.8.041033},
journal = {Physical Review. X},
number = 4,
volume = 8,
place = {United States},
year = {Tue Nov 27 00:00:00 EST 2018},
month = {Tue Nov 27 00:00:00 EST 2018}
}
https://doi.org/10.1103/PhysRevX.8.041033
Web of Science
Figures / Tables:
Works referenced in this record:
Mott transition in one dimension
journal, February 1997
- Giamarchi, T.
- Physica B: Condensed Matter, Vol. 230-232
Corner Transfer Matrix Renormalization Group Method
journal, April 1996
- Nishino, Tomotoshi; Okunishi, Kouichi
- Journal of the Physical Society of Japan, Vol. 65, Issue 4
Matrix product state renormalization
journal, November 2016
- Bal, M.; Rams, M. M.; Zauner, V.
- Physical Review B, Vol. 94, Issue 20
On form-factor expansions for the XXZ chain in the massive regime
journal, May 2015
- Dugave, Maxime; Göhmann, Frank; Kozlowski, Karol K.
- Journal of Statistical Mechanics: Theory and Experiment, Vol. 2015, Issue 5
Time-Dependent Variational Principle for Quantum Lattices
journal, August 2011
- Haegeman, Jutho; Cirac, J. Ignacio; Osborne, Tobias J.
- Physical Review Letters, Vol. 107, Issue 7
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
One-dimensional Bose-Hubbard model with nearest-neighbor interaction
journal, May 2000
- Kühner, Till D.; White, Steven R.; Monien, H.
- Physical Review B, Vol. 61, Issue 18
Hybrid-space density matrix renormalization group study of the doped two-dimensional Hubbard model
journal, March 2017
- Ehlers, G.; White, S. R.; Noack, R. M.
- Physical Review B, Vol. 95, Issue 12
Finite Correlation Length Scaling with Infinite Projected Entangled-Pair States
journal, July 2018
- Corboz, Philippe; Czarnik, Piotr; Kapteijns, Geert
- Physical Review X, Vol. 8, Issue 3
Vertical-Arrow Correlation Length in the Eight-Vertex Model and the Low-Lying Excitations of the Hamiltonian
journal, November 1973
- Johnson, James D.; Krinsky, Samuel; McCoy, Barry M.
- Physical Review A, Vol. 8, Issue 5
Gap scaling at Berezinskii-Kosterlitz-Thouless quantum critical points in one-dimensional Hubbard and Heisenberg models
journal, April 2015
- Dalmonte, M.; Carrasquilla, J.; Taddia, L.
- Physical Review B, Vol. 91, Issue 16
Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions
journal, July 2018
- Rader, Michael; Läuchli, Andreas M.
- Physical Review X, Vol. 8, Issue 3
The renormalization group: Critical phenomena and the Kondo problem
journal, October 1975
- Wilson, Kenneth G.
- Reviews of Modern Physics, Vol. 47, Issue 4
Variational tensor network renormalization in imaginary time: Two-dimensional quantum compass model at finite temperature
journal, May 2016
- Czarnik, Piotr; Dziarmaga, Jacek; Oleś, Andrzej M.
- Physical Review B, Vol. 93, Issue 18
Boson localization and the superfluid-insulator transition
journal, July 1989
- Fisher, Matthew P. A.; Weichman, Peter B.; Grinstein, G.
- Physical Review B, Vol. 40, Issue 1
Effective Harmonic-Fluid Approach to Low-Energy Properties of One-Dimensional Quantum Fluids
journal, December 1981
- Haldane, F. D. M.
- Physical Review Letters, Vol. 47, Issue 25
Matrix product states for critical spin chains: Finite-size versus finite-entanglement scaling
journal, August 2012
- Pirvu, B.; Vidal, G.; Verstraete, F.
- Physical Review B, Vol. 86, Issue 7
Error estimates for extrapolations with matrix-product states
journal, January 2018
- Hubig, C.; Haegeman, J.; Schollwöck, U.
- Physical Review B, Vol. 97, Issue 4
Density-matrix algorithms for quantum renormalization groups
journal, October 1993
- White, Steven R.
- Physical Review B, Vol. 48, Issue 14
Critical properties and Rényi entropies of the spin- chain
journal, April 2012
- Dalmonte, M.; Ercolessi, E.; Taddia, L.
- Physical Review B, Vol. 85, Issue 16
Critical behaviour of spin-s Heisenberg antiferromagnetic chains: analytic and numerical results
journal, March 1989
- Affleck, I.; Gepner, D.; Schulz, H. J.
- Journal of Physics A: Mathematical and General, Vol. 22, Issue 5
Variational approach to projected entangled pair states at finite temperature
journal, July 2015
- Czarnik, Piotr; Dziarmaga, Jacek
- Physical Review B, Vol. 92, Issue 3
Transfer matrices and excitations with matrix product states
journal, May 2015
- Zauner, V.; Draxler, D.; Vanderstraeten, L.
- New Journal of Physics, Vol. 17, Issue 5
Theory of Finite-Entanglement Scaling at One-Dimensional Quantum Critical Points
journal, June 2009
- Pollmann, Frank; Mukerjee, Subroto; Turner, Ari M.
- Physical Review Letters, Vol. 102, Issue 25
Corner Transfer Matrix Algorithm for Classical Renormalization Group
journal, October 1997
- Nishino, Tomotoshi; Okunishi, Kouichi
- Journal of the Physical Society of Japan, Vol. 66, Issue 10
Ornstein-Zernike theory for finite range Ising models above T c
journal, March 2003
- Campanino, Massimo; Ioffe, Dmitry; Velenik, Y. van
- Probability Theory and Related Fields, Vol. 125, Issue 3
A practical introduction to tensor networks: Matrix product states and projected entangled pair states
journal, October 2014
- Orús, Román
- Annals of Physics, Vol. 349
Conductivity of a Luttinger liquid
journal, October 1992
- Giamarchi, T.; Millis, A. J.
- Physical Review B, Vol. 46, Issue 15
Matrix product states represent ground states faithfully
journal, March 2006
- Verstraete, F.; Cirac, J. I.
- Physical Review B, Vol. 73, Issue 9
Competing States in the - Model: Uniform -Wave State versus Stripe State
journal, July 2014
- Corboz, Philippe; Rice, T. M.; Troyer, Matthias
- Physical Review Letters, Vol. 113, Issue 4
Critical theory of quantum spin chains
journal, October 1987
- Affleck, Ian; Haldane, F. D. M.
- Physical Review B, Vol. 36, Issue 10
Density matrix formulation for quantum renormalization groups
journal, November 1992
- White, Steven R.
- Physical Review Letters, Vol. 69, Issue 19
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
Unifying time evolution and optimization with matrix product states
journal, October 2016
- Haegeman, Jutho; Lubich, Christian; Oseledets, Ivan
- Physical Review B, Vol. 94, Issue 16
Phase diagram of the anisotropic spin-2 XXZ model: Infinite-system density matrix renormalization group study
journal, June 2013
- Kjäll, Jonas A.; Zaletel, Michael P.; Mong, Roger S. K.
- Physical Review B, Vol. 87, Issue 23
Thermal form factor approach to the ground-state correlation functions of the XXZ chain in the antiferromagnetic massive regime
journal, September 2016
- Dugave, Maxime; Göhmann, Frank; Kozlowski, Karol K.
- Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 39
Numerical renormalization group at criticality
journal, April 1996
- Nishino, T.; Okunishi, K.; Kikuchi, M.
- Physics Letters A, Vol. 213, Issue 1-2
Finitely correlated states on quantum spin chains
journal, March 1992
- Fannes, M.; Nachtergaele, B.; Werner, R. F.
- Communications in Mathematical Physics, Vol. 144, Issue 3
Critical Behavior of the Antiferromagnetic Heisenberg Chain
journal, June 1996
- Hallberg, K.; Wang, X. Q. G.; Horsch, P.
- Physical Review Letters, Vol. 76, Issue 26
Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms
journal, December 2015
- LeBlanc, J. P. F.; Antipov, Andrey E.; Becca, Federico
- Physical Review X, Vol. 5, Issue 4
Ultracold bosons with short-range interaction in regular optical lattices
journal, January 2016
- Krutitsky, Konstantin V.
- Physics Reports, Vol. 607
Relationship between d-Dimensional Quantal Spin Systems and (d+1)-Dimensional Ising Systems: Equivalence, Critical Exponents and Systematic Approximants of the Partition Function and Spin Correlations
journal, November 1976
- Suzuki, M.
- Progress of Theoretical Physics, Vol. 56, Issue 5
The density-matrix renormalization group in the age of matrix product states
journal, January 2011
- Schollwöck, Ulrich
- Annals of Physics, Vol. 326, Issue 1
Critical Behavior of Two-Dimensional Hydrogen-Bonded Antiferroelectrics
journal, June 1969
- Wu, F. Y.
- Physical Review Letters, Vol. 22, Issue 22
Artificial Spin-Ice and Vertex Models
journal, January 2017
- Cugliandolo, Leticia F.
- Journal of Statistical Physics, Vol. 167, Issue 3-4
Truncating an exact matrix product state for the XY model: Transfer matrix and its renormalization
journal, December 2015
- Rams, Marek M.; Zauner, Valentin; Bal, Matthias
- Physical Review B, Vol. 92, Issue 23
Numerical renormalization-group study of the correlation functions of the antiferromagnetic spin-1/2 Heisenberg chain
journal, July 1995
- Hallberg, Karen A.; Horsch, Peter; Martínez, Gerardo
- Physical Review B, Vol. 52, Issue 2
New Quantum Phase in a One-Dimensional Josephson Array
journal, November 1997
- Glazman, L. I.; Larkin, A. I.
- Physical Review Letters, Vol. 79, Issue 19
Accuracy of the density-matrix renormalization-group method
journal, June 1996
- Legeza, Örs; Fáth, Gábor
- Physical Review B, Vol. 53, Issue 21
Statistical Mechanics of the Model. II. Spin-Correlation Functions
journal, February 1971
- Barouch, Eytan; McCoy, Barry M.
- Physical Review A, Vol. 3, Issue 2
Critical behavior of the two-dimensional icosahedron model
journal, December 2017
- Ueda, Hiroshi; Okunishi, Kouichi; Krčmár, Roman
- Physical Review E, Vol. 96, Issue 6
Variational optimization algorithms for uniform matrix product states
journal, January 2018
- Zauner-Stauber, V.; Vanderstraeten, L.; Fishman, M. T.
- Physical Review B, Vol. 97, Issue 4
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
Relationship among Exactly Soluble Models of Critical Phenomena. I: 2D Ising Model, Dimer Problem and the Generalized
journal, November 1971
- Suzuki, Masuo
- Progress of Theoretical Physics, Vol. 46, Issue 5
Tensor network decompositions in the presence of a global symmetry
journal, November 2010
- Singh, Sukhwinder; Pfeifer, Robert N. C.; Vidal, Guifré
- Physical Review A, Vol. 82, Issue 5
The 16-vertex model and its even and odd 8-vertex subcases on the square lattice
journal, September 2017
- Assis, Michael
- Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 39
Density matrix renormalization group algorithms with a single center site
journal, November 2005
- White, Steven R.
- Physical Review B, Vol. 72, Issue 18
Scaling of the gap, fidelity susceptibility, and Bloch oscillations across the superfluid-to-Mott-insulator transition in the one-dimensional Bose-Hubbard model
journal, April 2013
- Carrasquilla, Juan; Manmana, Salvatore R.; Rigol, Marcos
- Physical Review A, Vol. 87, Issue 4
Residual entropies for three-dimensional frustrated spin systems with tensor networks
journal, October 2018
- Vanderstraeten, Laurens; Vanhecke, Bram; Verstraete, Frank
- Physical Review E, Vol. 98, Issue 4
Correlation functions of one-dimensional quantum systems
journal, March 1989
- Giamarchi, T.; Schulz, H. J.
- Physical Review B, Vol. 39, Issue 7
Spin-(1/2 antiferromagnetic XXZ chain: New results and insights
journal, February 1989
- Singh, Rajiv R. P.; Fisher, Michael E.; Shankar, R.
- Physical Review B, Vol. 39, Issue 4
Scaling of entanglement support for matrix product states
journal, July 2008
- Tagliacozzo, L.; de Oliveira, Thiago. R.; Iblisdir, S.
- Physical Review B, Vol. 78, Issue 2
Tensor network states and algorithms in the presence of a global U(1) symmetry
journal, March 2011
- Singh, Sukhwinder; Pfeifer, Robert N. C.; Vidal, Guifre
- Physical Review B, Vol. 83, Issue 11
Phase diagrams and correlation exponents for quantum spin chains of arbitrary spin quantum number
journal, November 1986
- Schulz, H. J.
- Physical Review B, Vol. 34, Issue 9
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
Critical properties of the eight-vertex model in a field
journal, September 2016
- Krčmár, Roman; Šamaj, Ladislav
- EPL (Europhysics Letters), Vol. 115, Issue 5
Conformal data from finite entanglement scaling
journal, January 2015
- Stojevic, Vid; Haegeman, Jutho; McCulloch, I. P.
- Physical Review B, Vol. 91, Issue 3
Ornstein-Zernike decay in the ground state of the quantum Ising model in a strong transverse field
journal, April 1991
- Kennedy, Tom
- Communications in Mathematical Physics, Vol. 137, Issue 3
Classical Simulation of Infinite-Size Quantum Lattice Systems in Two Spatial Dimensions
journal, December 2008
- Jordan, J.; Orús, R.; Vidal, G.
- Physical Review Letters, Vol. 101, Issue 25
Figures / Tables found in this record: