1
|
Dwivedi A, Lopez-Ruiz MA, Iyengar SS. Resource Optimization for Quantum Dynamics with Tensor Networks: Quantum and Classical Algorithms. J Phys Chem A 2024; 128:6774-6797. [PMID: 39101545 DOI: 10.1021/acs.jpca.4c03407] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/06/2024]
Abstract
The exponential scaling of the quantum degrees of freedom with the size of the system is one of the biggest challenges in computational chemistry and particularly in quantum dynamics. We present a tensor network approach for the time-evolution of the nuclear degrees of freedom of multiconfigurational chemical systems at a reduced storage and computational complexity. We also present quantum algorithms for the resultant dynamics. To preserve the compression advantage achieved via tensor network decompositions, we present an adaptive algorithm for the regularization of nonphysical bond dimensions, preventing the potentially exponential growth of these with time. While applicable to any quantum dynamical problem, our method is particularly valuable for dynamical simulations of nuclear chemical systems. Our algorithm is demonstrated using ab initio potentials obtained for a symmetric hydrogen-bonded system, namely, the protonated 2,2'-bipyridine, and compared to exact diagonalization numerical results.
Collapse
Affiliation(s)
- Anurag Dwivedi
- Department of Chemistry, Indiana University, Bloomington, Indiana 47405, United States
- Indiana University Quantum Science and Engineering Center (IU-QSEC), Indiana University, Bloomington, Indiana 47405, United States
| | - Miguel Angel Lopez-Ruiz
- Department of Chemistry, Indiana University, Bloomington, Indiana 47405, United States
- Indiana University Quantum Science and Engineering Center (IU-QSEC), Indiana University, Bloomington, Indiana 47405, United States
| | - Srinivasan S Iyengar
- Department of Chemistry, Indiana University, Bloomington, Indiana 47405, United States
- Indiana University Quantum Science and Engineering Center (IU-QSEC), Indiana University, Bloomington, Indiana 47405, United States
| |
Collapse
|
2
|
Jørgensen FK, Delcey MG, Hedegård ED. Perspective: multi-configurational methods in bio-inorganic chemistry. Phys Chem Chem Phys 2024; 26:17443-17455. [PMID: 38868993 DOI: 10.1039/d4cp01297f] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/14/2024]
Abstract
Transition metal ions play crucial roles in the structure and function of numerous proteins, contributing to essential biological processes such as catalysis, electron transfer, and oxygen binding. However, accurately modeling the electronic structure and properties of metalloproteins poses significant challenges due to the complex nature of their electronic configurations and strong correlation effects. Multiconfigurational quantum chemistry methods are, in principle, the most appropriate tools for addressing these challenges, offering the capability to capture the inherent multi-reference character and strong electron correlation present in bio-inorganic systems. Yet their computational cost has long hindered wider adoption, making methods such as density functional theory (DFT) the method of choice. However, advancements over the past decade have substantially alleviated this limitation, rendering multiconfigurational quantum chemistry methods more accessible and applicable to a wider range of bio-inorganic systems. In this perspective, we discuss some of these developments and how they have already been used to answer some of the most important questions in bio-inorganic chemistry. We also comment on ongoing developments in the field and how the future of the field may evolve.
Collapse
Affiliation(s)
- Frederik K Jørgensen
- Department of Physics, Chemistry, and Pharmacy, University of Southern Denmark, Campusvej 55, 5230 Odense, Denmark.
| | - Mickaël G Delcey
- Department of Chemistry, Lund University, Naturvetarvägen 14, 221 00 Lund, Sweden
| | - Erik D Hedegård
- Department of Physics, Chemistry, and Pharmacy, University of Southern Denmark, Campusvej 55, 5230 Odense, Denmark.
- Department of Chemistry, Lund University, Naturvetarvägen 14, 221 00 Lund, Sweden
| |
Collapse
|
3
|
Guo W, Wei TC. Tensor network methods for extracting conformal field theory data from fixed-point tensors and defect coarse graining. Phys Rev E 2024; 109:034111. [PMID: 38632738 DOI: 10.1103/physreve.109.034111] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2023] [Accepted: 12/01/2023] [Indexed: 04/19/2024]
Abstract
We present a comprehensive study on extracting conformal field theory data using tensor network methods, especially, from the fixed-point tensor of the linearized tensor renormalization group (lTRG) for the classical two-dimensional Ising model near the critical temperature. Utilizing two different methods, we extract operator scaling dimensions and operator product expansion coefficients by introducing defects on the lattice and by employing the fixed-point tensor. We also explore the effects of pointlike defects in the lattice on the coarse-graining process. We find that there is a correspondence between coarse-grained defect tensors and conformal states obtained from the lTRG fixed-point equation. We also analyze the capabilities and limitations of our proposed coarse-graining scheme for tensor networks with pointlike defects, including graph-independent local truncation (GILT) and higher-order tensor renormalization group (HOTRG). Our results provide a better understanding of the capacity and limitations of the tensor renormalization group scheme in coarse-graining defect tensors, and we show that GILT+HOTRG can be used to give accurate two- and four-point functions under specific conditions. We also find that employing the minimal canonical form further improves the stability of the RG flow.
Collapse
Affiliation(s)
- Wenhan Guo
- C. N. Yang Institute for Theoretical Physics and Department of Physics and Astronomy, State University of New York at Stony Brook, Stony Brook, New York 11794-3840, USA
| | - Tzu-Chieh Wei
- C. N. Yang Institute for Theoretical Physics and Department of Physics and Astronomy, State University of New York at Stony Brook, Stony Brook, New York 11794-3840, USA
| |
Collapse
|
4
|
Bilokon V, Bilokon E, Bañuls MC, Cichy A, Sotnikov A. Many-body correlations in one-dimensional optical lattices with alkaline-earth(-like) atoms. Sci Rep 2023; 13:9857. [PMID: 37330574 DOI: 10.1038/s41598-023-37077-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/24/2023] [Accepted: 06/15/2023] [Indexed: 06/19/2023] Open
Abstract
We explore the rich nature of correlations in the ground state of ultracold atoms trapped in state-dependent optical lattices. In particular, we consider interacting fermionic ytterbium or strontium atoms, realizing a two-orbital Hubbard model with two spin components. We analyze the model in one-dimensional setting with the experimentally relevant hierarchy of tunneling and interaction amplitudes by means of exact diagonalization and matrix product states approaches, and study the correlation functions in density, spin, and orbital sectors as functions of variable densities of atoms in the ground and metastable excited states. We show that in certain ranges of densities these atomic systems demonstrate strong density-wave, ferro- and antiferromagnetic, as well as antiferroorbital correlations.
Collapse
Affiliation(s)
- Valeriia Bilokon
- V. N. Karazin Kharkiv National University, Svobody Sq. 4, Kharkiv, 61022, Ukraine
- Adam Mickiewicz University in Poznań, Uniwersytetu Poznańskiego 2, 61-614, Poznań, Poland
| | - Elvira Bilokon
- V. N. Karazin Kharkiv National University, Svobody Sq. 4, Kharkiv, 61022, Ukraine
- Adam Mickiewicz University in Poznań, Uniwersytetu Poznańskiego 2, 61-614, Poznań, Poland
| | - Mari Carmen Bañuls
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748, Garching, Germany
- Munich Centre for Quantum Science and Technology (MCQST), Schellingstrasse 4, 80799, Munich, Germany
| | - Agnieszka Cichy
- Adam Mickiewicz University in Poznań, Uniwersytetu Poznańskiego 2, 61-614, Poznań, Poland
- Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55099, Mainz, Germany
| | - Andrii Sotnikov
- V. N. Karazin Kharkiv National University, Svobody Sq. 4, Kharkiv, 61022, Ukraine.
- Kharkiv Institute of Physics and Technology, Akademichna 1, Kharkiv, 61108, Ukraine.
| |
Collapse
|
5
|
Gleis A, Li JW, von Delft J. Controlled Bond Expansion for Density Matrix Renormalization Group Ground State Search at Single-Site Costs. PHYSICAL REVIEW LETTERS 2023; 130:246402. [PMID: 37390431 DOI: 10.1103/physrevlett.130.246402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/10/2022] [Revised: 03/24/2023] [Accepted: 04/07/2023] [Indexed: 07/02/2023]
Abstract
DMRG ground state search algorithms employing symmetries must be able to expand virtual bond spaces by adding or changing symmetry sectors if these lower the energy. Traditional single-site DMRG does not allow bond expansion; two-site DMRG does, but at much higher computational costs. We present a controlled bond expansion (CBE) algorithm that yields two-site accuracy and convergence per sweep, at single-site costs. Given a matrix product state Ψ defining a variational space, CBE identifies parts of the orthogonal space carrying significant weight in HΨ and expands bonds to include only these. CBE-DMRG uses no mixing parameters and is fully variational. Using CBE-DMRG, we show that the Kondo-Heisenberg model on a width 4 cylinder features two distinct phases differing in their Fermi surface volumes.
Collapse
Affiliation(s)
- Andreas Gleis
- Arnold Sommerfeld Center for Theoretical Physics, Center for NanoScience, and Munich Center for Quantum Science and Technology, Ludwig-Maximilians-Universität München, 80333 Munich, Germany
| | - Jheng-Wei Li
- Arnold Sommerfeld Center for Theoretical Physics, Center for NanoScience, and Munich Center for Quantum Science and Technology, Ludwig-Maximilians-Universität München, 80333 Munich, Germany
| | - Jan von Delft
- Arnold Sommerfeld Center for Theoretical Physics, Center for NanoScience, and Munich Center for Quantum Science and Technology, Ludwig-Maximilians-Universität München, 80333 Munich, Germany
| |
Collapse
|
6
|
Yu XJ, Ding C, Xu L. Quantum criticality of a Z_{3}-symmetric spin chain with long-range interactions. Phys Rev E 2023; 107:054122. [PMID: 37329095 DOI: 10.1103/physreve.107.054122] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2023] [Accepted: 04/28/2023] [Indexed: 06/18/2023]
Abstract
Based on large-scale density matrix renormalization group techniques, we investigate the critical behaviors of quantum three-state Potts chains with long-range interactions. Using fidelity susceptibility as an indicator, we obtain a complete phase diagram of the system. The results show that as the long-range interaction power α increases, the critical points f_{c}^{*} shift towards lower values. In addition, the critical threshold α_{c}(≈1.43) of the long-range interaction power is obtained for the first time by a nonperturbative numerical method. This indicates that the critical behavior of the system can be naturally divided into two distinct universality classes, namely the long-range (α<α_{c}) and short-range (α>α_{c}) universality classes, qualitatively consistent with the classical ϕ^{3} effective field theory. This work provides a useful reference for further research on phase transitions in quantum spin chains with long-range interaction.
Collapse
Affiliation(s)
- Xue-Jia Yu
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
| | - Chengxiang Ding
- School of Science and Engineering of Mathematics and Physics, Anhui University of Technology, Maanshan, Anhui 243002, China
| | - Limei Xu
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Collaborative Innovation Center of Quantum Matter, Beijing, China
- Interdisciplinary Institute of Light-Element Quantum Materials and Research Center for Light-Element Advanced Materials, Peking University, Beijing, China
| |
Collapse
|
7
|
Zhou HQ, Shi QQ, Dai YW. Fidelity Mechanics: Analogues of the Four Thermodynamic Laws and Landauer's Principle. ENTROPY (BASEL, SWITZERLAND) 2022; 24:1306. [PMID: 36141191 PMCID: PMC9498036 DOI: 10.3390/e24091306] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/21/2022] [Revised: 08/04/2022] [Accepted: 08/22/2022] [Indexed: 06/16/2023]
Abstract
Fidelity mechanics is formalized as a framework for investigating critical phenomena in quantum many-body systems. Fidelity temperature is introduced for quantifying quantum fluctuations, which, together with fidelity entropy and fidelity internal energy, constitute three basic state functions in fidelity mechanics, thus enabling us to formulate analogues of the four thermodynamic laws and Landauer's principle at zero temperature. Fidelity flows, which are irreversible, are defined and may be interpreted as an alternative form of renormalization group flows. Thus, fidelity mechanics offers a means to characterize both stable and unstable fixed points: divergent fidelity temperature for unstable fixed points and zero-fidelity temperature and (locally) maximal fidelity entropy for stable fixed points. In addition, fidelity entropy behaves differently at an unstable fixed point for topological phase transitions and at a stable fixed point for topological quantum states of matter. A detailed analysis of fidelity mechanical-state functions is presented for six fundamental models-the quantum spin-1/2 XY model, the transverse-field quantum Ising model in a longitudinal field, the quantum spin-1/2 XYZ model, the quantum spin-1/2 XXZ model in a magnetic field, the quantum spin-1 XYZ model, and the spin-1/2 Kitaev model on a honeycomb lattice for illustrative purposes. We also present an argument to justify why the thermodynamic, psychological/computational, and cosmological arrows of time should align with each other, with the psychological/computational arrow of time being singled out as a master arrow of time.
Collapse
Affiliation(s)
- Huan-Qiang Zhou
- Centre for Modern Physics, Chongqing University, Chongqing 400044, China
| | | | | |
Collapse
|
8
|
Hoyer CE, Hu H, Lu L, Knecht S, Li X. Relativistic Kramers-Unrestricted Exact-Two-Component Density Matrix Renormalization Group. J Phys Chem A 2022; 126:5011-5020. [PMID: 35881436 DOI: 10.1021/acs.jpca.2c02150] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
In this work we develop a variational relativistic density matrix renormalization group (DMRG) approach within the exact-two-component (X2C) framework (X2C-DMRG), using spinor orbitals optimized with the two-component relativistic complete active space self-consistent field. We investigate fine-structure splittings of p- (Ga, In, Tl) and d-block (Sc, Y, La) atoms and excitation energies of monohydride molecules (GeH, SnH, and TlH) with X2C-DMRG calculations using an all-electron relativistic Hamiltonian in a Kramers-unrestricted basis. We find that X2C-DMRG yields accurate 2P and 2D splittings compared to multireference configuration interaction with singles and doubles (MRCISD). We also investigated the degree of symmetry breaking in the atomic multiplets and convergence of electron correlation in the total energies. Symmetry breaking can be large in some cases (∼30 meV); however, increasing the number of renormalized block states m for the DMRG optimization recovers the symmetry breaking by several orders of magnitude. Encouragingly, we find the convergence of electron correlation to be close to MRCISDTQ5 quality. Relativistic X2C-DMRG approaches are important for cases where spin-orbit coupling is significant and the underlying reference wave function requires a large determinantal space. We are able to obtain quantitatively correct fine-structure splittings for systems up to 1019 number of determinants with traditional CI approaches, which are currently unfeasible to converge for the field.
Collapse
Affiliation(s)
- Chad E Hoyer
- Department of Chemistry, University of Washington, Seattle, Washington 98195, United States
| | - Hang Hu
- Department of Chemistry, University of Washington, Seattle, Washington 98195, United States
| | - Lixin Lu
- Department of Chemistry, University of Washington, Seattle, Washington 98195, United States
| | - Stefan Knecht
- Algorithmiq Ltd., Kanavakatu 3C, FI-00160 Helsinki, Finland.,Abteilung SHE Chemie, GSI Helmholtzzentrum für Schwerionenforschung, DE-64291 Darmstadt, Germany.,Department Chemie, Johannes-Gutenberg Universität Mainz, DE-55128 Mainz, Germany
| | - Xiaosong Li
- Department of Chemistry, University of Washington, Seattle, Washington 98195, United States
| |
Collapse
|
9
|
Chen SYC, Huang CM, Hsing CW, Goan HS, Kao YJ. Variational quantum reinforcement learning via evolutionary optimization. MACHINE LEARNING: SCIENCE AND TECHNOLOGY 2022. [DOI: 10.1088/2632-2153/ac4559] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Abstract
Recent advances in classical reinforcement learning (RL) and quantum computation point to a promising direction for performing RL on a quantum computer. However, potential applications in quantum RL are limited by the number of qubits available in modern quantum devices. Here, we present two frameworks for deep quantum RL tasks using gradient-free evolutionary optimization. First, we apply the amplitude encoding scheme to the Cart-Pole problem, where we demonstrate the quantum advantage of parameter saving using amplitude encoding. Second, we propose a hybrid framework where the quantum RL agents are equipped with a hybrid tensor network-variational quantum circuit (TN-VQC) architecture to handle inputs of dimensions exceeding the number of qubits. This allows us to perform quantum RL in the MiniGrid environment with 147-dimensional inputs. The hybrid TN-VQC architecture provides a natural way to perform efficient compression of the input dimension, enabling further quantum RL applications on noisy intermediate-scale quantum devices.
Collapse
|
10
|
Montangero S, Rico E, Silvi P. Loop-free tensor networks for high-energy physics. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2022; 380:20210065. [PMID: 34923837 DOI: 10.1098/rsta.2021.0065] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2021] [Accepted: 09/13/2021] [Indexed: 06/14/2023]
Abstract
This brief review introduces the reader to tensor network methods, a powerful theoretical and numerical paradigm spawning from condensed matter physics and quantum information science and increasingly exploited in different fields of research, from artificial intelligence to quantum chemistry. Here, we specialize our presentation on the application of loop-free tensor network methods to the study of high-energy physics problems and, in particular, to the study of lattice gauge theories where tensor networks can be applied in regimes where Monte Carlo methods are hindered by the sign problem. This article is part of the theme issue 'Quantum technologies in particle physics'.
Collapse
Affiliation(s)
- Simone Montangero
- Dipartimento di Fisica e Astronomia 'G. Galilei', Università di Padova, Padova 35131, Italy
- Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Padova, Padova 35131, Italy
- Padua Quantum Technologies Research Center, Università degli Studi di Padova, Padova, 35131, Italy
| | - Enrique Rico
- Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, Bilbao 48080, Spain
- IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, Bilbao 48009, Spain
| | - Pietro Silvi
- Center for Quantum Physics, and Institute for Experimental Physics, University of Innsbruck, Innsbruck 6020, Austria
| |
Collapse
|
11
|
Zhou C, Hermes MR, Wu D, Bao JJ, Pandharkar R, King DS, Zhang D, Scott TR, Lykhin AO, Gagliardi L, Truhlar DG. Electronic structure of strongly correlated systems: recent developments in multiconfiguration pair-density functional theory and multiconfiguration nonclassical-energy functional theory. Chem Sci 2022; 13:7685-7706. [PMID: 35865899 PMCID: PMC9261488 DOI: 10.1039/d2sc01022d] [Citation(s) in RCA: 14] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2022] [Accepted: 06/01/2022] [Indexed: 12/25/2022] Open
Abstract
Strong electron correlation plays an important role in transition-metal and heavy-metal chemistry, magnetic molecules, bond breaking, biradicals, excited states, and many functional materials, but it provides a significant challenge for modern electronic structure theory. The treatment of strongly correlated systems usually requires a multireference method to adequately describe spin densities and near-degeneracy correlation. However, quantitative computation of dynamic correlation with multireference wave functions is often difficult or impractical. Multiconfiguration pair-density functional theory (MC-PDFT) provides a way to blend multiconfiguration wave function theory and density functional theory to quantitatively treat both near-degeneracy correlation and dynamic correlation in strongly correlated systems; it is more affordable than multireference perturbation theory, multireference configuration interaction, or multireference coupled cluster theory and more accurate for many properties than Kohn–Sham density functional theory. This perspective article provides a brief introduction to strongly correlated systems and previously reviewed progress on MC-PDFT followed by a discussion of several recent developments and applications of MC-PDFT and related methods, including localized-active-space MC-PDFT, generalized active-space MC-PDFT, density-matrix-renormalization-group MC-PDFT, hybrid MC-PDFT, multistate MC-PDFT, spin–orbit coupling, analytic gradients, and dipole moments. We also review the more recently introduced multiconfiguration nonclassical-energy functional theory (MC-NEFT), which is like MC-PDFT but allows for other ingredients in the nonclassical-energy functional. We discuss two new kinds of MC-NEFT methods, namely multiconfiguration density coherence functional theory and machine-learned functionals. This feature article overviews recent work on active spaces, matrix product reference states, treatment of quasidegeneracy, hybrid theory, density-coherence functionals, machine-learned functionals, spin–orbit coupling, gradients, and dipole moments.![]()
Collapse
Affiliation(s)
- Chen Zhou
- Department of Chemistry, Chemical Theory Center, Minnesota Supercomputing Institute, University of Minnesota, 207 Pleasant Street SE, Minneapolis, MN 55455-0431, USA
| | - Matthew R. Hermes
- Department of Chemistry, Pritzker School of Molecular Engineering, The James Franck Institute and Chicago Center for Theoretical Chemistry, The University of Chicago, Chicago, IL 60637, USA
| | - Dihua Wu
- Department of Chemistry, Chemical Theory Center, Minnesota Supercomputing Institute, University of Minnesota, 207 Pleasant Street SE, Minneapolis, MN 55455-0431, USA
| | - Jie J. Bao
- Department of Chemistry, Chemical Theory Center, Minnesota Supercomputing Institute, University of Minnesota, 207 Pleasant Street SE, Minneapolis, MN 55455-0431, USA
| | - Riddhish Pandharkar
- Department of Chemistry, Pritzker School of Molecular Engineering, The James Franck Institute and Chicago Center for Theoretical Chemistry, The University of Chicago, Chicago, IL 60637, USA
- Argonne National Laboratory, Lemont, Illinois 60439, USA
| | - Daniel S. King
- Department of Chemistry, Pritzker School of Molecular Engineering, The James Franck Institute and Chicago Center for Theoretical Chemistry, The University of Chicago, Chicago, IL 60637, USA
| | - Dayou Zhang
- Department of Chemistry, Chemical Theory Center, Minnesota Supercomputing Institute, University of Minnesota, 207 Pleasant Street SE, Minneapolis, MN 55455-0431, USA
| | - Thais R. Scott
- Department of Chemistry, Pritzker School of Molecular Engineering, The James Franck Institute and Chicago Center for Theoretical Chemistry, The University of Chicago, Chicago, IL 60637, USA
| | - Aleksandr O. Lykhin
- Department of Chemistry, Pritzker School of Molecular Engineering, The James Franck Institute and Chicago Center for Theoretical Chemistry, The University of Chicago, Chicago, IL 60637, USA
| | - Laura Gagliardi
- Department of Chemistry, Pritzker School of Molecular Engineering, The James Franck Institute and Chicago Center for Theoretical Chemistry, The University of Chicago, Chicago, IL 60637, USA
- Argonne National Laboratory, Lemont, Illinois 60439, USA
| | - Donald G. Truhlar
- Department of Chemistry, Chemical Theory Center, Minnesota Supercomputing Institute, University of Minnesota, 207 Pleasant Street SE, Minneapolis, MN 55455-0431, USA
| |
Collapse
|
12
|
Causer L, Bañuls MC, Garrahan JP. Optimal sampling of dynamical large deviations via matrix product states. Phys Rev E 2021; 103:062144. [PMID: 34271638 DOI: 10.1103/physreve.103.062144] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/01/2021] [Accepted: 05/07/2021] [Indexed: 12/23/2022]
Abstract
The large deviation statistics of dynamical observables is encoded in the spectral properties of deformed Markov generators. Recent works have shown that tensor network methods are well suited to compute accurately the relevant leading eigenvalues and eigenvectors. However, the efficient generation of the corresponding rare trajectories is a harder task. Here, we show how to exploit the matrix product state approximation of the dominant eigenvector to implement an efficient sampling scheme which closely resembles the optimal (so-called "Doob") dynamics that realizes the rare events. We demonstrate our approach on three well-studied lattice models, the Fredrickson-Andersen and East kinetically constrained models, and the symmetric simple exclusion process. We discuss how to generalize our approach to higher dimensions.
Collapse
Affiliation(s)
- Luke Causer
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom.,Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - Mari Carmen Bañuls
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany.,Munich Center for Quantum Science and Technology (MCQST), Schellingstrasse 4, D-80799 München, Germany
| | - Juan P Garrahan
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom.,Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom.,All Souls College, Oxford, United Kingdom
| |
Collapse
|
13
|
Causer L, Lesanovsky I, Bañuls MC, Garrahan JP. Dynamics and large deviation transitions of the XOR-Fredrickson-Andersen kinetically constrained model. Phys Rev E 2020; 102:052132. [PMID: 33327088 DOI: 10.1103/physreve.102.052132] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/24/2020] [Accepted: 11/09/2020] [Indexed: 11/07/2022]
Abstract
We study a one-dimensional classical stochastic kinetically constrained model (KCM) inspired by Rydberg atoms in their "facilitated" regime, where sites can flip only if a single of their nearest neighbors is excited. We call this model "XOR-FA" to distinguish it from the standard Fredrickson-Andersen (FA) model. We describe the dynamics of the XOR-FA model, including its relation to simple exclusion processes in its domain wall representation. The interesting relaxation dynamics of the XOR-FA is related to the prominence of large dynamical fluctuations that lead to phase transitions between active and inactive dynamical phases as in other KCMs. By means of numerical tensor network methods we study in detail such transitions in the dynamical large deviation regime.
Collapse
Affiliation(s)
- Luke Causer
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom.,Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - Igor Lesanovsky
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom.,Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom.,Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
| | - Mari Carmen Bañuls
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany.,Munich Center for Quantum Science and Technology (MCQST), Schellingstrasse 4, D-80799 München, Germany
| | - Juan P Garrahan
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom.,Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| |
Collapse
|
14
|
Carmen Bañuls M, Cichy K. Review on novel methods for lattice gauge theories. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2020; 83:024401. [PMID: 31846938 DOI: 10.1088/1361-6633/ab6311] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
Formulating gauge theories on a lattice offers a genuinely non-perturbative way of studying quantum field theories, and has led to impressive achievements. In particular, it significantly deepened our understanding of quantum chromodynamics. Yet, some very relevant problems remain inherently challenging, such as real time evolution, or the presence of a chemical potential, cases in which Monte Carlo simulations are hindered by a sign problem. In the last few years, a number of possible alternatives have been put forward, based on quantum information ideas, which could potentially open the access to areas of research that have so far eluded more standard methods. They include tensor network calculations, quantum simulations with different physical platforms and quantum computations, and constitute nowadays a vibrant research area. Experts from different fields, including experimental and theoretical high energy physics, condensed matter, and quantum information, are turning their attention to these interdisciplinary possibilities, and driving the progress of the field. The aim of this article is to review the status and perspectives of these new avenues for the exploration of lattice gauge theories.
Collapse
Affiliation(s)
- Mari Carmen Bañuls
- Max-Planck Institut, für Quantenoptik, Garching 85748, Germany. Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, Munich 80799, Germany
| | | |
Collapse
|
15
|
Vanhecke B, Haegeman J, Van Acoleyen K, Vanderstraeten L, Verstraete F. Scaling Hypothesis for Matrix Product States. PHYSICAL REVIEW LETTERS 2019; 123:250604. [PMID: 31922783 DOI: 10.1103/physrevlett.123.250604] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/12/2019] [Indexed: 06/10/2023]
Abstract
We study critical spin systems and field theories using matrix product states, and formulate a scaling hypothesis in terms of operators, eigenvalues of the transfer matrix, and lattice spacing in the case of field theories. The critical point, exponents, and central charge are determined by optimizing them to obtain a data collapse. We benchmark this method by studying critical Ising and Potts models, where we also obtain a scaling Ansatz for the correlation length and entanglement entropy. The formulation of those scaling functions turns out to be crucial for studying critical quantum field theories on the lattice. For the case of λϕ^{4} with mass parameter μ^{2} and lattice spacing a, we demonstrate a double data collapse for the correlation length δξ(μ,λ,D)=ξ[over ˜]((α-α_{c})(δ/a)^{-1/ν}) with D the bond dimension, δ the gap between eigenvalues of the transfer matrix, and α_{c}=μ_{R}^{2}/λ the parameter which fixes the critical quantum field theory.
Collapse
Affiliation(s)
- Bram Vanhecke
- Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium
| | - Jutho Haegeman
- Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium
| | - Karel Van Acoleyen
- Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium
| | - Laurens Vanderstraeten
- Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium
| | - Frank Verstraete
- Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium
| |
Collapse
|
16
|
Bañuls MC, Garrahan JP. Using Matrix Product States to Study the Dynamical Large Deviations of Kinetically Constrained Models. PHYSICAL REVIEW LETTERS 2019; 123:200601. [PMID: 31809110 DOI: 10.1103/physrevlett.123.200601] [Citation(s) in RCA: 25] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2019] [Indexed: 06/10/2023]
Abstract
Here we demonstrate that tensor network techniques-originally devised for the analysis of quantum many-body problems-are well suited for the detailed study of rare event statistics in kinetically constrained models (KCMs). As concrete examples, we consider the Fredrickson-Andersen and East models, two paradigmatic KCMs relevant to the modeling of glasses. We show how variational matrix product states allow us to numerically approximate-systematically and with high accuracy-the leading eigenstates of the tilted dynamical generators, which encode the large deviation statistics of the dynamics. Via this approach, we can study system sizes beyond what is possible with other methods, allowing us to characterize in detail the finite size scaling of the trajectory-space phase transition of these models, the behavior of spectral gaps, and the spatial structure and "entanglement" properties of dynamical phases. We discuss the broader implications of our results.
Collapse
Affiliation(s)
- Mari Carmen Bañuls
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany
- Munich Center for Quantum Science and Technology (MCQST), Schellingstrasse 4, D-80799 München, Germany
| | - Juan P Garrahan
- School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kindom
- Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
| |
Collapse
|
17
|
Varizi AD, Drumond RC. Quantum Ising model in a period-2 modulated transverse field. Phys Rev E 2019; 100:022104. [PMID: 31574608 DOI: 10.1103/physreve.100.022104] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2019] [Indexed: 11/07/2022]
Abstract
We study a finite spin-1/2 Ising chain with a spatially alternating transverse field of period 2. By means of a Jordan-Wigner transformation for even and odd sites, we are able to map it into a one-dimensional model of free fermions. We determine the ground-state energies in the positive- and negative-parity subspaces (subspaces with an even or odd total number of down spins, respectively) and compare them in order to establish the ground-state energy for the entire Hamiltonian. We derive closed-form expressions for this energy gap between the different parity subspaces and analyze its behavior and dependence on the system size in the various regimes of the applied field. Finally, we suggest an expression for the correlation length of such a model that is consistent with the various values found in the literature for its behavior in the vicinity of critical points.
Collapse
Affiliation(s)
- Adalberto D Varizi
- Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais 31270-901, Brazil
| | - Raphael C Drumond
- Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais 31270-901, Brazil
| |
Collapse
|
18
|
Fdez. Galván I, Vacher M, Alavi A, Angeli C, Aquilante F, Autschbach J, Bao JJ, Bokarev SI, Bogdanov NA, Carlson RK, Chibotaru LF, Creutzberg J, Dattani N, Delcey MG, Dong SS, Dreuw A, Freitag L, Frutos LM, Gagliardi L, Gendron F, Giussani A, González L, Grell G, Guo M, Hoyer CE, Johansson M, Keller S, Knecht S, Kovačević G, Källman E, Li Manni G, Lundberg M, Ma Y, Mai S, Malhado JP, Malmqvist PÅ, Marquetand P, Mewes SA, Norell J, Olivucci M, Oppel M, Phung QM, Pierloot K, Plasser F, Reiher M, Sand AM, Schapiro I, Sharma P, Stein CJ, Sørensen LK, Truhlar DG, Ugandi M, Ungur L, Valentini A, Vancoillie S, Veryazov V, Weser O, Wesołowski TA, Widmark PO, Wouters S, Zech A, Zobel JP, Lindh R. OpenMolcas: From Source Code to Insight. J Chem Theory Comput 2019; 15:5925-5964. [DOI: 10.1021/acs.jctc.9b00532] [Citation(s) in RCA: 399] [Impact Index Per Article: 79.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
Affiliation(s)
- Ignacio Fdez. Galván
- Department of Chemistry − Ångström Laboratory, Uppsala University, P.O. Box 538, SE-751 21 Uppsala, Sweden
- Department of Chemistry − BMC, Uppsala University, P.O. Box 576, SE-751 23 Uppsala, Sweden
| | - Morgane Vacher
- Department of Chemistry − Ångström Laboratory, Uppsala University, P.O. Box 538, SE-751 21 Uppsala, Sweden
| | - Ali Alavi
- Max Planck Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany
| | - Celestino Angeli
- Dipartimento di Scienze Chimiche e Farmaceutiche, Università di Ferrara, Via Luigi Borsari 46, 44121 Ferrara, Italy
| | - Francesco Aquilante
- Département de Chimie Physique, Université de Genève, 30 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland
| | - Jochen Autschbach
- Department of Chemistry, University at Buffalo, State University of New York, Buffalo, New York 14260-3000, United States
| | - Jie J. Bao
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Sergey I. Bokarev
- Institut für Physik, Universität Rostock, Albert-Einstein-Straße 23-24, 18059 Rostock, Germany
| | - Nikolay A. Bogdanov
- Max Planck Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany
| | - Rebecca K. Carlson
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Liviu F. Chibotaru
- Theory of Nanomaterials Group, University of Leuven, Celestijnenlaan 200F, Leuven 3001, Belgium
| | - Joel Creutzberg
- Department of Physics, AlbaNova University Center, Stockholm University, SE-106 91 Stockholm, Sweden
- Division of Theoretical Chemistry, Kemicentrum, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden
| | - Nike Dattani
- Harvard Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, United States
| | - Mickaël G. Delcey
- Department of Chemistry − Ångström Laboratory, Uppsala University, P.O. Box 538, SE-751 21 Uppsala, Sweden
| | - Sijia S. Dong
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Andreas Dreuw
- Interdisciplinary Center for Scientific Computing, Heidelberg University, Im Neuenheimer Feld 205 A, 69120 Heidelberg, Germany
| | - Leon Freitag
- Laboratorium für Physikalische Chemie, ETH Zürich, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Luis Manuel Frutos
- Departamento de Química Analítica, Química Física e Ingeniería Química, and Instituto de Investigación Química “Andrés M. del Río”, Universidad de Alcalá, E-28871 Alcalá de Henares, Madrid, Spain
| | - Laura Gagliardi
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Frédéric Gendron
- Department of Chemistry, University at Buffalo, State University of New York, Buffalo, New York 14260-3000, United States
| | - Angelo Giussani
- Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, United Kingdom
- Instituto de Ciencia Molecular, Universitat de València, Apartado 22085, ES-46071 Valencia, Spain
| | - Leticia González
- Institute of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Straße 17, 1090 Vienna, Austria
| | - Gilbert Grell
- Institut für Physik, Universität Rostock, Albert-Einstein-Straße 23-24, 18059 Rostock, Germany
| | - Meiyuan Guo
- Department of Chemistry − Ångström Laboratory, Uppsala University, P.O. Box 538, SE-751 21 Uppsala, Sweden
| | - Chad E. Hoyer
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Marcus Johansson
- Division of Theoretical Chemistry, Kemicentrum, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden
| | - Sebastian Keller
- Laboratorium für Physikalische Chemie, ETH Zürich, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Stefan Knecht
- Laboratorium für Physikalische Chemie, ETH Zürich, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Goran Kovačević
- Division of Materials Physics, Ruđer Bošković Institute, P.O.B. 180, Bijenička 54, HR-10002 Zagreb, Croatia
| | - Erik Källman
- Department of Chemistry − Ångström Laboratory, Uppsala University, P.O. Box 538, SE-751 21 Uppsala, Sweden
| | - Giovanni Li Manni
- Max Planck Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany
| | - Marcus Lundberg
- Department of Chemistry − Ångström Laboratory, Uppsala University, P.O. Box 538, SE-751 21 Uppsala, Sweden
| | - Yingjin Ma
- Laboratorium für Physikalische Chemie, ETH Zürich, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Sebastian Mai
- Institute of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Straße 17, 1090 Vienna, Austria
| | - João Pedro Malhado
- Department of Chemistry, Imperial College London, London SW7 2AZ, United Kingdom
| | - Per Åke Malmqvist
- Division of Theoretical Chemistry, Kemicentrum, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden
| | - Philipp Marquetand
- Institute of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Straße 17, 1090 Vienna, Austria
| | - Stefanie A. Mewes
- Interdisciplinary Center for Scientific Computing, Heidelberg University, Im Neuenheimer Feld 205 A, 69120 Heidelberg, Germany
- Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Advanced Study (NZIAS), Massey University Albany, Private Bag
102904, Auckland 0632, New Zealand
| | - Jesper Norell
- Department of Physics, AlbaNova University Center, Stockholm University, SE-106 91 Stockholm, Sweden
| | - Massimo Olivucci
- Department of Biotechnology, Chemistry and Pharmacy, University of Siena, via A. Moro 2, 53100 Siena, Italy
- Department of Chemistry, Bowling Green State University, Bowling Green, Ohio 43403, United States
- USIAS and Institut de Physique et Chimie des Matériaux de Strasbourg, Université de Strasbourg-CNRS, 67034 Strasbourg, France
| | - Markus Oppel
- Institute of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Straße 17, 1090 Vienna, Austria
| | - Quan Manh Phung
- Department of Chemistry, KU Leuven, Celestijnenlaan 200F, Leuven 3001, Belgium
| | - Kristine Pierloot
- Department of Chemistry, KU Leuven, Celestijnenlaan 200F, Leuven 3001, Belgium
| | - Felix Plasser
- Department of Chemistry, Loughborough University, Loughborough LE11 3TU, United Kingdom
| | - Markus Reiher
- Laboratorium für Physikalische Chemie, ETH Zürich, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Andrew M. Sand
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Igor Schapiro
- Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel
| | - Prachi Sharma
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Christopher J. Stein
- Laboratorium für Physikalische Chemie, ETH Zürich, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Lasse Kragh Sørensen
- Department of Chemistry − Ångström Laboratory, Uppsala University, P.O. Box 538, SE-751 21 Uppsala, Sweden
| | - Donald G. Truhlar
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Mihkel Ugandi
- Department of Chemistry − Ångström Laboratory, Uppsala University, P.O. Box 538, SE-751 21 Uppsala, Sweden
| | - Liviu Ungur
- Department of Chemistry, National University of Singapore, 117543 Singapore
| | - Alessio Valentini
- Theoretical Physical Chemistry, Research Unit MolSys, Allée du 6 Août, 11, 4000 Liège, Belgium
| | - Steven Vancoillie
- Division of Theoretical Chemistry, Kemicentrum, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden
| | - Valera Veryazov
- Division of Theoretical Chemistry, Kemicentrum, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden
| | - Oskar Weser
- Max Planck Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany
| | - Tomasz A. Wesołowski
- Département de Chimie Physique, Université de Genève, 30 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland
| | - Per-Olof Widmark
- Division of Theoretical Chemistry, Kemicentrum, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden
| | - Sebastian Wouters
- Brantsandpatents, Pauline van Pottelsberghelaan 24, 9051 Sint-Denijs-Westrem, Belgium
| | - Alexander Zech
- Département de Chimie Physique, Université de Genève, 30 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland
| | - J. Patrick Zobel
- Division of Theoretical Chemistry, Kemicentrum, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden
| | - Roland Lindh
- Department of Chemistry − BMC, Uppsala University, P.O. Box 576, SE-751 23 Uppsala, Sweden
- Uppsala Center for Computational Chemistry (UC3), Uppsala University, P.O. Box 596, SE-751 24 Uppsala, Sweden
| |
Collapse
|
19
|
Li ZH. Ground states of long-range interacting fermions in one spatial dimension. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2019; 31:255601. [PMID: 30870818 DOI: 10.1088/1361-648x/ab0fcf] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
We systematically explore the ground state properties of one dimensional fermions with long-range interactions decaying in a power law ∼[Formula: see text] through the density matrix renormalization group algorithm. By comparing values of Luttinger liquid parameters precisely measured in two different ways, we show convincing evidence that Luttinger liquid theory is valid if [Formula: see text] is larger than some threshold, otherwise the theory breaks down. Combining analysis on structure factor, charge gap and charge stiffness, we determine how the metal-insulator transition point develops as the interaction range is continuously tuned. A region in the range of [Formula: see text] has small interactions and finite charge gaps, but, interestingly, it shows metallic nature at the same time. We obtain approximate phase diagrams for the entire parameter space and for band fillings equal to 1/2 and 1/3. Finally, we compare certain bosonization and field theory formulas with our quasi-exact numerical results, from which disagreements are found.
Collapse
Affiliation(s)
- Zhi-Hua Li
- School of Science, Xi'an Technological University, Xi'an 710021, People's Republic of China
| |
Collapse
|
20
|
Frahm LH, Pfannkuche D. Ultrafast ab Initio Quantum Chemistry Using Matrix Product States. J Chem Theory Comput 2019; 15:2154-2165. [DOI: 10.1021/acs.jctc.8b01291] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Lars-Hendrik Frahm
- I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany
| | - Daniela Pfannkuche
- I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany
| |
Collapse
|
21
|
Yang C, Binder FC, Narasimhachar V, Gu M. Matrix Product States for Quantum Stochastic Modeling. PHYSICAL REVIEW LETTERS 2018; 121:260602. [PMID: 30636154 DOI: 10.1103/physrevlett.121.260602] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/20/2018] [Revised: 08/10/2018] [Indexed: 06/09/2023]
Abstract
In stochastic modeling, there has been a significant effort towards finding predictive models that predict a stochastic process' future using minimal information from its past. Meanwhile, in condensed matter physics, matrix product states (MPS) are known as a particularly efficient representation of 1D spin chains. In this Letter, we associate each stochastic process with a suitable quantum state of a spin chain. We then show that the optimal predictive model for the process leads directly to an MPS representation of the associated quantum state. Conversely, MPS methods offer a systematic construction of the best known quantum predictive models. This connection allows an improved method for computing the quantum memory needed for generating optimal predictions. We prove that this memory coincides with the entanglement of the associated spin chain across the past-future bipartition.
Collapse
Affiliation(s)
- Chengran Yang
- School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore, Singapore
- Complexity institute, Nanyang Technological University, 639798 Singapore, Singapore
| | - Felix C Binder
- School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore, Singapore
- Complexity institute, Nanyang Technological University, 639798 Singapore, Singapore
| | - Varun Narasimhachar
- School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore, Singapore
- Complexity institute, Nanyang Technological University, 639798 Singapore, Singapore
| | - Mile Gu
- School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore, Singapore
- Complexity institute, Nanyang Technological University, 639798 Singapore, Singapore
- Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore
| |
Collapse
|
22
|
Zou Y, Milsted A, Vidal G. Conformal Data and Renormalization Group Flow in Critical Quantum Spin Chains Using Periodic Uniform Matrix Product States. PHYSICAL REVIEW LETTERS 2018; 121:230402. [PMID: 30576172 DOI: 10.1103/physrevlett.121.230402] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/22/2018] [Revised: 08/08/2018] [Indexed: 06/09/2023]
Abstract
We establish that a Bloch-state ansatz based on periodic uniform matrix product states (PUMPS), originally designed to capture single-quasiparticle excitations in gapped systems, is in fact capable of accurately approximating all low-energy eigenstates of critical quantum spin chains on the circle. When combined with the methods of [Milsted and Vidal, Phys. Rev. B 96, 245105PRBMDO2469-995010.1103/PhysRevB.96.245105] based on the Koo-Saleur formula, PUMPS Bloch states can then be used to identify each low-energy eigenstate of a chain made of up to hundreds of spins with its corresponding scaling operator in the emergent conformal field theory (CFT). This enables the following two tasks that we demonstrate using the quantum Ising model and a recently proposed generalization thereof due to O'Brien and Fendley [Phys. Rev. Lett. 120, 206403]. (i) From the spectrum of low energies and momenta we extract conformal data (specifying the emergent CFT) with unprecedented numerical accuracy. (ii) By changing the lattice size, we investigate nonperturbatively the renormalization group flow of the low-energy spectrum between two CFTs. In our example, where the flow is from the tricritical Ising CFT to the Ising CFT, we obtain excellent agreement with an analytical result [Klassen and Melzer, Nucl. Phys. B370, 51110.1016/0550-3213(92)90422-8] conjectured to describe the flow of the first spectral gap directly in the continuum.
Collapse
Affiliation(s)
- Yijian Zou
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
- University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - Ashley Milsted
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
| | - Guifre Vidal
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
| |
Collapse
|
23
|
Breaking the curse of dimension for the electronic Schrödinger equation with functional analysis. COMPUT THEOR CHEM 2018. [DOI: 10.1016/j.comptc.2018.08.017] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
|
24
|
Móller NS, de Paula AL, Drumond RC. Shielding property for thermal equilibrium states in the quantum Ising model. Phys Rev E 2018; 97:032101. [PMID: 29776175 DOI: 10.1103/physreve.97.032101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/04/2017] [Indexed: 11/07/2022]
Abstract
We show that Gibbs states of nonhomogeneous transverse Ising chains satisfy a shielding property. Namely, whatever the fields on each spin and exchange couplings between neighboring spins are, if the field in one particular site is null, then the reduced states of the subchains to the right and to the left of this site are exactly the Gibbs states of each subchain alone. Therefore, even if there is a strong exchange coupling between the extremal sites of each subchain, the Gibbs states of the each subchain behave as if there is no interaction between them. In general, if a lattice can be divided into two disconnected regions separated by an interface of sites with zero applied field, then we can guarantee a similar result only if the surface contains a single site. Already for an interface with two sites we show an example where the property does not hold. When it holds, however, we show that if a perturbation of the Hamiltonian parameters is done in one side of the lattice, then the other side is completely unchanged, with regard to both its equilibrium state and dynamics.
Collapse
Affiliation(s)
- N S Móller
- Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil
| | - A L de Paula
- Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil
| | - R C Drumond
- Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil
| |
Collapse
|
25
|
Battaglia S, Keller S, Knecht S. Efficient Relativistic Density-Matrix Renormalization Group Implementation in a Matrix-Product Formulation. J Chem Theory Comput 2018; 14:2353-2369. [DOI: 10.1021/acs.jctc.7b01065] [Citation(s) in RCA: 30] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Stefano Battaglia
- ETH Zürich, Laboratorium für Physikalische Chemie, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Sebastian Keller
- ETH Zürich, Laboratorium für Physikalische Chemie, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Stefan Knecht
- ETH Zürich, Laboratorium für Physikalische Chemie, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| |
Collapse
|
26
|
The acidic pH-induced structural changes in apo-CP43 by spectral methodologies and molecular dynamics simulations. J Mol Struct 2018. [DOI: 10.1016/j.molstruc.2017.09.082] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
|
27
|
Yanai T, Saitow M, Xiong XG, Chalupský J, Kurashige Y, Guo S, Sharma S. Multistate Complete-Active-Space Second-Order Perturbation Theory Based on Density Matrix Renormalization Group Reference States. J Chem Theory Comput 2017; 13:4829-4840. [DOI: 10.1021/acs.jctc.7b00735] [Citation(s) in RCA: 43] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/28/2022]
Affiliation(s)
- Takeshi Yanai
- Department
of Theoretical and Computational Molecular Science, Institute for Molecular Science, Okazaki, 444-8585 Aichi Japan
- The Graduate University for Advanced Studies, Myodaiji, Okazaki, Aichi 444-8585, Japan
| | - Masaaki Saitow
- The Graduate University for Advanced Studies, Myodaiji, Okazaki, Aichi 444-8585, Japan
| | - Xiao-Gen Xiong
- Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China
| | - Jakub Chalupský
- Institute
of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic, Flemingovo nám. 2, 16110 Prague 6, Czech Republic
| | - Yuki Kurashige
- Department
of Chemistry, Graduate School of Science, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyou-ku, Kyoto 606-8520, Japan
- Japan Science and Technology Agency, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
| | - Sheng Guo
- Division
of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, United States
| | - Sandeep Sharma
- Department
of Chemistry and Biochemistry, University of Colorado, Boulder, Boulder, Colorado 80302, United States
| |
Collapse
|
28
|
Li Z, Chan GKL. Spin-Projected Matrix Product States: Versatile Tool for Strongly Correlated Systems. J Chem Theory Comput 2017; 13:2681-2695. [PMID: 28467847 DOI: 10.1021/acs.jctc.7b00270] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
We present a new wave function ansatz that combines the strengths of spin projection with the language of matrix product states (MPS) and matrix product operators (MPO) as used in the density matrix renormalization group (DMRG). Specifically, spin-projected matrix product states (SP-MPS) are constructed as [Formula: see text], where [Formula: see text] is the spin projector for total spin S and |ΨMPS(N,M)⟩ is an MPS wave function with a given particle number N and spin projection M. This new ansatz possesses several attractive features: (1) It provides a much simpler route to achieve spin adaptation (i.e., to create eigenfunctions of Ŝ2) compared to explicitly incorporating the non-Abelian SU(2) symmetry into the MPS. In particular, since the underlying state |ΨMPS(N,M)⟩ in the SP-MPS uses only Abelian symmetries, one does not need the singlet embedding scheme for nonsinglet states, as normally employed in spin-adapted DMRG, to achieve a single consistent variationally optimized state. (2) Due to the use of |ΨMPS(N,M)⟩ as its underlying state, the SP-MPS can be closely connected to broken-symmetry mean-field states. This allows one to straightforwardly generate the large number of broken-symmetry guesses needed to explore complex electronic landscapes in magnetic systems. Further, this connection can be exploited in the future development of quantum embedding theories for open-shell systems. (3) The sum of MPOs representation for the Hamiltonian and spin projector [Formula: see text] naturally leads to an embarrassingly parallel algorithm for computing expectation values and optimizing SP-MPS. (4) Optimizing SP-MPS belongs to the variation-after-projection (VAP) class of spin-projected theories. Unlike usual spin-projected theories based on determinants, the SP-MPS ansatz can be made essentially exact simply by increasing the bond dimensions in |ΨMPS(N,M)⟩. Computing excited states is also simple by imposing orthogonality constraints, which are simple to implement with MPS. To illustrate the versatility of SP-MPS, we formulate algorithms for the optimization of ground and excited states, develop perturbation theory based on SP-MPS, and describe how to evaluate spin-independent and spin-dependent properties such as the reduced density matrices. We demonstrate the numerical performance of SP-MPS with applications to several models typical of strong correlation, including the Hubbard model, and [2Fe-2S] and [4Fe-4S] model complexes.
Collapse
Affiliation(s)
- Zhendong Li
- Division of Chemistry and Chemical Engineering, California Institute of Technology , Pasadena, California 91125, United States
| | - Garnet Kin-Lic Chan
- Division of Chemistry and Chemical Engineering, California Institute of Technology , Pasadena, California 91125, United States
| |
Collapse
|
29
|
Bañuls MC, Cichy K, Ignacio Cirac J, Jansen K, Kühn S, Saito H. Towards overcoming the Monte Carlo sign problem with tensor networks. EPJ WEB OF CONFERENCES 2017. [DOI: 10.1051/epjconf/201713704001] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
30
|
Yang S, Gu ZC, Wen XG. Loop Optimization for Tensor Network Renormalization. PHYSICAL REVIEW LETTERS 2017; 118:110504. [PMID: 28368642 DOI: 10.1103/physrevlett.118.110504] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/01/2017] [Indexed: 06/07/2023]
Abstract
We introduce a tensor renormalization group scheme for coarse graining a two-dimensional tensor network that can be successfully applied to both classical and quantum systems on and off criticality. The key innovation in our scheme is to deform a 2D tensor network into small loops and then optimize the tensors on each loop. In this way, we remove short-range entanglement at each iteration step and significantly improve the accuracy and stability of the renormalization flow. We demonstrate our algorithm in the classical Ising model and a frustrated 2D quantum model.
Collapse
Affiliation(s)
- Shuo Yang
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
| | - Zheng-Cheng Gu
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
- Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
| | - Xiao-Gang Wen
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| |
Collapse
|
31
|
Mei Z, Bolech CJ. Derivation of matrix product states for the Heisenberg spin chain with open boundary conditions. Phys Rev E 2017; 95:032127. [PMID: 28415216 DOI: 10.1103/physreve.95.032127] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/23/2016] [Indexed: 11/07/2022]
Abstract
Using the algebraic Bethe Ansatz, we derive a matrix product representation of the exact Bethe-Ansatz states of the six-vertex Heisenberg chain (either XXX or XXZ and spin-1/2) with open boundary conditions. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices that act on a tensor product of auxiliary spaces. As compared to the matrix product states of the same Heisenberg chain but with periodic boundary conditions, the dimension of the exact auxiliary matrices is enlarged as if the conserved number of spin-flips considered would have been doubled. This result is generic for any non-nested integrable model, as is clear from our derivation, and we further show this by providing an additional example of the same matrix product state construction for a well-known model of a gas of interacting bosons. Counterintuitively, the matrices do not depend on the spatial coordinate despite the open boundaries, and thus they suggest generic ways of exploiting (emergent) translational invariance both for finite size and in the thermodynamic limit.
Collapse
Affiliation(s)
- Zhongtao Mei
- Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221-0011, USA
| | - C J Bolech
- Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221-0011, USA
| |
Collapse
|
32
|
Dey D, Maiti D, Kumar M. An efficient density matrix renormalization group algorithm for chains with periodic boundary condition. PAPERS IN PHYSICS 2016. [DOI: 10.4279/pip.080006] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022] Open
Abstract
The Density Matrix Renormalization Group (DMRG) is a state-of-the-art numerical technique for a one dimensional quantum many-body system; but calculating accurate results for a system with Periodic Boundary Condition (PBC) from the conventional DMRG has been a challenging job from the inception of DMRG. The recent development of the Matrix Product State (MPS) algorithm gives a new approach to find accurate results for the one dimensional PBC system. The most efficient implementation of the MPS algorithm can scale as O($p \times m^3$), where $p$ can vary from 4 to $m^2$. In this paper, we propose a new DMRG algorithm, which is very similar to the conventional DMRG and gives comparable accuracy to that of MPS. The computation effort of the new algorithm goes as O($m^3$) and the conventional DMRG code can be easily modified for the new algorithm.Received: 2 August 2016, Accepted: 12 October 2016; Edited by: K. Hallberg; DOI: http://dx.doi.org/10.4279/PIP.080006Cite as: D Dey, D Maiti, M Kumar, Papers in Physics 8, 080006 (2016)This paper, by D Dey, D Maiti, M Kumar, is licensed under the Creative Commons Attribution License 3.0.
Collapse
|
33
|
Krumnow C, Veis L, Legeza Ö, Eisert J. Fermionic Orbital Optimization in Tensor Network States. PHYSICAL REVIEW LETTERS 2016; 117:210402. [PMID: 27911544 DOI: 10.1103/physrevlett.117.210402] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2015] [Indexed: 06/06/2023]
Abstract
Tensor network states and specifically matrix-product states have proven to be a powerful tool for simulating ground states of strongly correlated spin models. Recently, they have also been applied to interacting fermionic problems, specifically in the context of quantum chemistry. A new freedom arising in such nonlocal fermionic systems is the choice of orbitals, it being far from clear what choice of fermionic orbitals to make. In this Letter, we propose a way to overcome this challenge. We suggest a method intertwining the optimization over matrix product states with suitable fermionic Gaussian mode transformations. The described algorithm generalizes basis changes in the spirit of the Hartree-Fock method to matrix-product states, and provides a black box tool for basis optimization in tensor network methods.
Collapse
Affiliation(s)
- C Krumnow
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
| | - L Veis
- Strongly Correlated Systems "Lendület" Research Group, Wigner Research Centre for Physics, Hungarian Academy of Sciences, 1525 Budapest, Hungary
- J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, 18223 Prague, Czech Republic
| | - Ö Legeza
- Strongly Correlated Systems "Lendület" Research Group, Wigner Research Centre for Physics, Hungarian Academy of Sciences, 1525 Budapest, Hungary
| | - J Eisert
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
| |
Collapse
|
34
|
Keller S, Reiher M. Spin-adapted matrix product states and operators. J Chem Phys 2016; 144:134101. [DOI: 10.1063/1.4944921] [Citation(s) in RCA: 58] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Sebastian Keller
- ETH Zürich, Laboratory of Physical Chemistry, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Markus Reiher
- ETH Zürich, Laboratory of Physical Chemistry, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| |
Collapse
|
35
|
Keller S, Dolfi M, Troyer M, Reiher M. An efficient matrix product operator representation of the quantum chemical Hamiltonian. J Chem Phys 2015; 143:244118. [DOI: 10.1063/1.4939000] [Citation(s) in RCA: 106] [Impact Index Per Article: 11.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/01/2023] Open
|
36
|
Saitow M, Kurashige Y, Yanai T. Fully Internally Contracted Multireference Configuration Interaction Theory Using Density Matrix Renormalization Group: A Reduced-Scaling Implementation Derived by Computer-Aided Tensor Factorization. J Chem Theory Comput 2015; 11:5120-31. [DOI: 10.1021/acs.jctc.5b00270] [Citation(s) in RCA: 59] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Masaaki Saitow
- The Graduate University for Advanced Studies, Myodaiji, Okazaki, Aichi 444-8585, Japan
| | - Yuki Kurashige
- The Graduate University for Advanced Studies, Myodaiji, Okazaki, Aichi 444-8585, Japan
- Department
of Theoretical and Computational Molecular Science, Institute for Molecular Science, Okazaki, Aichi 444-8585, Japan
- Japan Science and Technology Agency, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
| | - Takeshi Yanai
- The Graduate University for Advanced Studies, Myodaiji, Okazaki, Aichi 444-8585, Japan
- Department
of Theoretical and Computational Molecular Science, Institute for Molecular Science, Okazaki, Aichi 444-8585, Japan
| |
Collapse
|
37
|
Liu W, Wang C, Li Y, Lao Y, Han Y, Guo GC, Zhao YH, He L. Replica exchange molecular dynamics optimization of tensor network states for quantum many-body systems. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:085601. [PMID: 25654245 DOI: 10.1088/0953-8984/27/8/085601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Tensor network states (TNS) methods combined with the Monte Carlo (MC) technique have been proven a powerful algorithm for simulating quantum many-body systems. However, because the ground state energy is a highly non-linear function of the tensors, it is easy to get stuck in local minima when optimizing the TNS of the simulated physical systems. To overcome this difficulty, we introduce a replica-exchange molecular dynamics optimization algorithm to obtain the TNS ground state, based on the MC sampling technique, by mapping the energy function of the TNS to that of a classical mechanical system. The method is expected to effectively avoid local minima. We make benchmark tests on a 1D Hubbard model based on matrix product states (MPS) and a Heisenberg J1-J2 model on square lattice based on string bond states (SBS). The results show that the optimization method is robust and efficient compared to the existing results.
Collapse
Affiliation(s)
- Wenyuan Liu
- Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China. Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | | | | | | | | | | | | | | |
Collapse
|
38
|
Universal order parameters and quantum phase transitions: a finite-size approach. Sci Rep 2015; 5:7673. [PMID: 25567585 PMCID: PMC4286789 DOI: 10.1038/srep07673] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2014] [Accepted: 12/05/2014] [Indexed: 11/29/2022] Open
Abstract
We propose a method to construct universal order parameters for quantum phase transitions in many-body lattice systems. The method exploits the H-orthogonality of a few near-degenerate lowest states of the Hamiltonian describing a given finite-size system, which makes it possible to perform finite-size scaling and take full advantage of currently available numerical algorithms. An explicit connection is established between the fidelity per site between two H-orthogonal states and the energy gap between the ground state and low-lying excited states in the finite-size system. The physical information encoded in this gap arising from finite-size fluctuations clarifies the origin of the universal order parameter. We demonstrate the procedure for the one-dimensional quantum formulation of the q-state Potts model, for q = 2, 3, 4 and 5, as prototypical examples, using finite-size data obtained from the density matrix renormalization group algorithm.
Collapse
|
39
|
Khoromskaia V, Khoromskij BN. Tensor numerical methods in quantum chemistry: from Hartree–Fock to excitation energies. Phys Chem Chem Phys 2015; 17:31491-509. [DOI: 10.1039/c5cp01215e] [Citation(s) in RCA: 41] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/31/2022]
Abstract
We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations.
Collapse
Affiliation(s)
- Venera Khoromskaia
- Max-Planck-Institute for Dynamics of Complex Systems
- Magdeburg
- Germany
- Max-Planck Institute for Mathematics in the Sciences
- Leipzig
| | | |
Collapse
|
40
|
Kurashige Y, Chalupský J, Lan TN, Yanai T. Complete active space second-order perturbation theory with cumulant approximation for extended active-space wavefunction from density matrix renormalization group. J Chem Phys 2014; 141:174111. [DOI: 10.1063/1.4900878] [Citation(s) in RCA: 100] [Impact Index Per Article: 10.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Yuki Kurashige
- Department of Theoretical and Computational Molecular Science,
Institute for Molecular Science, Okazaki, Aichi 444-8585,
Japan
- The Graduate University for Advanced Studies, Myodaiji, Okazaki, Aichi 444-8585, Japan
- Japan Science and Technology Agency,
PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
| | - Jakub Chalupský
- Department of Theoretical and Computational Molecular Science,
Institute for Molecular Science, Okazaki, Aichi 444-8585,
Japan
| | - Tran Nguyen Lan
- The Graduate University for Advanced Studies, Myodaiji, Okazaki, Aichi 444-8585, Japan
| | - Takeshi Yanai
- Department of Theoretical and Computational Molecular Science,
Institute for Molecular Science, Okazaki, Aichi 444-8585,
Japan
- The Graduate University for Advanced Studies, Myodaiji, Okazaki, Aichi 444-8585, Japan
| |
Collapse
|
41
|
Cominotti M, Rossini D, Rizzi M, Hekking F, Minguzzi A. Optimal persistent currents for interacting bosons on a ring with a gauge field. PHYSICAL REVIEW LETTERS 2014; 113:025301. [PMID: 25062201 DOI: 10.1103/physrevlett.113.025301] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2013] [Indexed: 06/03/2023]
Abstract
We study persistent currents for interacting one-dimensional bosons on a tight ring trap, subjected to a rotating barrier potential, which induces an artificial U(1) gauge field. We show that, at intermediate interactions, the persistent current response is maximal, due to a subtle interplay of effects due to the barrier, the interaction, and quantum fluctuations. These results are relevant for ongoing experiments with ultracold atomic gases on mesoscopic rings.
Collapse
Affiliation(s)
- Marco Cominotti
- Université Grenoble Alpes, LPMMC, F-38000 Grenoble, France and CNRS, LPMMC, F-38000 Grenoble, France
| | - Davide Rossini
- NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy
| | - Matteo Rizzi
- Institut für Physik, Johannes-Gutenberg-Universität Mainz, Staudingerweg 7, D-55099 Mainz, Germany
| | - Frank Hekking
- Université Grenoble Alpes, LPMMC, F-38000 Grenoble, France and CNRS, LPMMC, F-38000 Grenoble, France
| | - Anna Minguzzi
- Université Grenoble Alpes, LPMMC, F-38000 Grenoble, France and CNRS, LPMMC, F-38000 Grenoble, France
| |
Collapse
|
42
|
Dai YW, Cho SY, Batchelor MT, Zhou HQ. Degenerate ground states and multiple bifurcations in a two-dimensional q-state quantum Potts model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:062142. [PMID: 25019759 DOI: 10.1103/physreve.89.062142] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2014] [Indexed: 06/03/2023]
Abstract
We numerically investigate the two-dimensional q-state quantum Potts model on the infinite square lattice by using the infinite projected entangled-pair state (iPEPS) algorithm. We show that the quantum fidelity, defined as an overlap measurement between an arbitrary reference state and the iPEPS ground state of the system, can detect q-fold degenerate ground states for the Z_{q} broken-symmetry phase. Accordingly, a multiple bifurcation of the quantum ground-state fidelity is shown to occur as the transverse magnetic field varies from the symmetry phase to the broken-symmetry phase, which means that a multiple-bifurcation point corresponds to a critical point. A (dis)continuous behavior of quantum fidelity at phase transition points characterizes a (dis)continuous phase transition. Similar to the characteristic behavior of the quantum fidelity, the magnetizations, as order parameters, obtained from the degenerate ground states exhibit multiple bifurcation at critical points. Each order parameter is also explicitly demonstrated to transform under the Z_{q} subgroup of the symmetry group of the Hamiltonian. We find that the q-state quantum Potts model on the square lattice undergoes a discontinuous (first-order) phase transition for q=3 and q=4 and a continuous phase transition for q=2 (the two-dimensional quantum transverse Ising model).
Collapse
Affiliation(s)
- Yan-Wei Dai
- Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, People's Republic of China
| | - Sam Young Cho
- Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, People's Republic of China
| | - Murray T Batchelor
- Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, People's Republic of China and Department of Theoretical Physics and Mathematical Science Institute, Australian National University, Canberra, ACT 0200, Australia
| | - Huan-Qiang Zhou
- Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, People's Republic of China
| |
Collapse
|
43
|
Kazeev V, Khammash M, Nip M, Schwab C. Direct solution of the Chemical Master Equation using quantized tensor trains. PLoS Comput Biol 2014; 10:e1003359. [PMID: 24626049 PMCID: PMC3953644 DOI: 10.1371/journal.pcbi.1003359] [Citation(s) in RCA: 44] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2013] [Accepted: 10/09/2013] [Indexed: 11/24/2022] Open
Abstract
The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements in the number of problem dimensions. We present a novel approach that has the potential to “lift” this curse of dimensionality. The approach is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for the low parametric, numerical representation of tensors. The QTT decomposition admits both, algorithms for basic tensor arithmetics with complexity scaling linearly in the dimension (number of species) and sub-linearly in the mode size (maximum copy number), and a numerical tensor rounding procedure which is stable and quasi-optimal. We show how the CME can be represented in QTT format, then use the exponentially-converging -discontinuous Galerkin discretization in time to reduce the CME evolution problem to a set of QTT-structured linear equations to be solved at each time step using an algorithm based on Density Matrix Renormalization Group (DMRG) methods from quantum chemistry. Our method automatically adapts the “basis” of the solution at every time step guaranteeing that it is large enough to capture the dynamics of interest but no larger than necessary, as this would increase the computational complexity. Our approach is demonstrated by applying it to three different examples from systems biology: independent birth-death process, an example of enzymatic futile cycle, and a stochastic switch model. The numerical results on these examples demonstrate that the proposed QTT method achieves dramatic speedups and several orders of magnitude storage savings over direct approaches. Stochastic models of chemical networks are necessary to quantitatively describe random fluctuations and other probabilistic phenomena within living cells. The Chemical Master Equation (CME) describes the time evolution of molecular abundance probabilities in these models, and is a basis for many stochastic simulation and analysis methods. Yet the CME is difficult to solve directly except for very simple structures. Indeed current approaches are susceptible to the curse of dimensionality, that is, the exponential growth of memory and computational requirements in the number of problem dimensions. In this paper, we propose a novel approach that has the potential to overcome the curse of dimensionality. It is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for numerical representation of tensors, using algorithms for basic tensor arithmetics with complexity scaling linearly in the number of reacting species considered, and sub-linearly in the maximum allowed copy number per species. We present this approach and demonstrate its effectiveness by applying it to three problems from systems biology. Numerical experiments are reported which show that several orders of magnitude memory savings is typically afforded by the new approach presented here.
Collapse
Affiliation(s)
- Vladimir Kazeev
- Seminar für Angewandte Mathematik, ETH Zürich, Zürich, Switzerland
| | - Mustafa Khammash
- Department of Biosystems Science and Engineering, ETH Zürich, Basel, Switzerland
- * E-mail:
| | - Michael Nip
- Department of Mechanical Engineering, UC Santa Barbara, Santa Barbara, California, United States of America
| | - Christoph Schwab
- Seminar für Angewandte Mathematik, ETH Zürich, Zürich, Switzerland
| |
Collapse
|
44
|
Matrix Product State Algorithms: DMRG, TEBD and Relatives. ACTA ACUST UNITED AC 2013. [DOI: 10.1007/978-3-642-35106-8_3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/20/2023]
|
45
|
Liu JH, Shi QQ, Wang HL, Links J, Zhou HQ. Universal construction of order parameters for translation-invariant quantum lattice systems with symmetry-breaking order. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:020102. [PMID: 23005705 DOI: 10.1103/physreve.86.020102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/02/2012] [Indexed: 06/01/2023]
Abstract
For any translation-invariant quantum lattice system with a symmetry group G, we propose a practical and universal construction of order parameters which identify quantum phase transitions with symmetry-breaking order. They are defined in terms of the fidelity between a ground state and its symmetry-transformed counterpart, and are computed through tensor network representations of the ground-state wave function. To illustrate our scheme, we consider three quantum systems on an infinite lattice in one spatial dimension, namely, the quantum Ising model in a transverse magnetic field, the quantum spin-1/2XYX model in an external magnetic field, and the quantum spin-1 XXZ model with single-ion anisotropy. All these models have symmetry group Z(2) and exhibit broken-symmetry phases. We also discuss the role of the order parameters in identifying factorized states.
Collapse
Affiliation(s)
- Jin-Hua Liu
- Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, People's Republic of China
| | | | | | | | | |
Collapse
|
46
|
Chan GKL. Low entanglement wavefunctions. WILEY INTERDISCIPLINARY REVIEWS-COMPUTATIONAL MOLECULAR SCIENCE 2012. [DOI: 10.1002/wcms.1095] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/30/2022]
|
47
|
Pižorn I, Verstraete F. Variational numerical renormalization group: bridging the gap between NRG and density matrix renormalization group. PHYSICAL REVIEW LETTERS 2012; 108:067202. [PMID: 22401115 DOI: 10.1103/physrevlett.108.067202] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2011] [Revised: 12/13/2011] [Indexed: 05/31/2023]
Abstract
The numerical renormalization group (NRG) is rephrased as a variational method with the cost function given by the sum of all the energies of the effective low-energy Hamiltonian. This allows us to systematically improve the spectrum obtained by NRG through sweeping. The ensuing algorithm has a lot of similarities to the density matrix renormalization group (DMRG) when targeting many states, and this synergy of NRG and DMRG combines the best of both worlds and extends their applicability. We illustrate this approach with simulations of a quantum spin chain and a single impurity Anderson model where the accuracy of the effective eigenstates is greatly enhanced as compared to the NRG, especially in the transition to the continuum limit.
Collapse
Affiliation(s)
- Iztok Pižorn
- University of Vienna, Faculty of Physics, Wien, Austria
| | | |
Collapse
|
48
|
Sasaki A, Kojo M, Hirose K, Goto H. Real-space finite-difference approach for multi-body systems: path-integral renormalization group method and direct energy minimization method. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2011; 23:434001. [PMID: 21998159 DOI: 10.1088/0953-8984/23/43/434001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
The path-integral renormalization group and direct energy minimization method of practical first-principles electronic structure calculations for multi-body systems within the framework of the real-space finite-difference scheme are introduced. These two methods can handle higher dimensional systems with consideration of the correlation effect. Furthermore, they can be easily extended to the multicomponent quantum systems which contain more than two kinds of quantum particles. The key to the present methods is employing linear combinations of nonorthogonal Slater determinants (SDs) as multi-body wavefunctions. As one of the noticeable results, the same accuracy as the variational Monte Carlo method is achieved with a few SDs. This enables us to study the entire ground state consisting of electrons and nuclei without the need to use the Born-Oppenheimer approximation. Recent activities on methodological developments aiming towards practical calculations such as the implementation of auxiliary field for Coulombic interaction, the treatment of the kinetic operator in imaginary-time evolutions, the time-saving double-grid technique for bare-Coulomb atomic potentials and the optimization scheme for minimizing the total-energy functional are also introduced. As test examples, the total energy of the hydrogen molecule, the atomic configuration of the methylene and the electronic structures of two-dimensional quantum dots are calculated, and the accuracy, availability and possibility of the present methods are demonstrated.
Collapse
Affiliation(s)
- Akira Sasaki
- Department of Precision Science and Technology, Graduate School of Engineering, Osaka University, Suita, Osaka, Japan
| | | | | | | |
Collapse
|
49
|
Kurashige Y, Yanai T. Second-order perturbation theory with a density matrix renormalization group self-consistent field reference function: Theory and application to the study of chromium dimer. J Chem Phys 2011; 135:094104. [DOI: 10.1063/1.3629454] [Citation(s) in RCA: 239] [Impact Index Per Article: 18.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022] Open
|
50
|
Schollwöck U. The density-matrix renormalization group: a short introduction. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2011; 369:2643-2661. [PMID: 21646271 DOI: 10.1098/rsta.2010.0382] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
The density-matrix renormalization group (DMRG) method has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. The DMRG is a method that shares features of a renormalization group procedure (which here generates a flow in the space of reduced density operators) and of a variational method that operates on a highly interesting class of quantum states, so-called matrix product states (MPSs). The DMRG method is presented here entirely in the MPS language. While the DMRG generally fails in larger two-dimensional systems, the MPS picture suggests a straightforward generalization to higher dimensions in the framework of tensor network states. The resulting algorithms, however, suffer from difficulties absent in one dimension, apart from a much more unfavourable efficiency, such that their ultimate success remains far from clear at the moment.
Collapse
Affiliation(s)
- Ulrich Schollwöck
- Department of Physics, Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, University of Munich, Theresienstrasse 37, 80333 Munich, Germany.
| |
Collapse
|