1
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Vlaar PCG, Corboz P. Efficient Tensor Network Algorithm for Layered Systems. PHYSICAL REVIEW LETTERS 2023; 130:130601. [PMID: 37067308 DOI: 10.1103/physrevlett.130.130601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2022] [Revised: 12/31/2022] [Accepted: 02/21/2023] [Indexed: 06/19/2023]
Abstract
Strongly correlated layered 2D systems are of central importance in condensed matter physics, but their numerical study is very challenging. Motivated by the enormous successes of tensor networks for 1D and 2D systems, we develop an efficient tensor network approach based on infinite projected entangled-pair states for layered 2D systems. Starting from an anisotropic 3D infinite projected entangled-pair state ansatz, we propose a contraction scheme in which the weakly interacting layers are effectively decoupled away from the center of the layers, such that they can be efficiently contracted using 2D contraction methods while keeping the center of the layers connected in order to capture the most relevant interlayer correlations. We present benchmark data for the anisotropic 3D Heisenberg model on a cubic lattice, which shows close agreement with quantum Monte Carlo and full 3D contraction results. Finally, we study the dimer to Néel phase transition in the Shastry-Sutherland model with interlayer coupling, a frustrated spin model that is out of reach of quantum Monte Carlo due to the negative sign problem.
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Affiliation(s)
- Patrick C G Vlaar
- Institute for Theoretical Physics and Delta Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
| | - Philippe Corboz
- Institute for Theoretical Physics and Delta Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
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2
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Sajjan M, Li J, Selvarajan R, Sureshbabu SH, Kale SS, Gupta R, Singh V, Kais S. Quantum machine learning for chemistry and physics. Chem Soc Rev 2022; 51:6475-6573. [PMID: 35849066 DOI: 10.1039/d2cs00203e] [Citation(s) in RCA: 14] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
Abstract
Machine learning (ML) has emerged as a formidable force for identifying hidden but pertinent patterns within a given data set with the objective of subsequent generation of automated predictive behavior. In recent years, it is safe to conclude that ML and its close cousin, deep learning (DL), have ushered in unprecedented developments in all areas of physical sciences, especially chemistry. Not only classical variants of ML, even those trainable on near-term quantum hardwares have been developed with promising outcomes. Such algorithms have revolutionized materials design and performance of photovoltaics, electronic structure calculations of ground and excited states of correlated matter, computation of force-fields and potential energy surfaces informing chemical reaction dynamics, reactivity inspired rational strategies of drug designing and even classification of phases of matter with accurate identification of emergent criticality. In this review we shall explicate a subset of such topics and delineate the contributions made by both classical and quantum computing enhanced machine learning algorithms over the past few years. We shall not only present a brief overview of the well-known techniques but also highlight their learning strategies using statistical physical insight. The objective of the review is not only to foster exposition of the aforesaid techniques but also to empower and promote cross-pollination among future research in all areas of chemistry which can benefit from ML and in turn can potentially accelerate the growth of such algorithms.
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Affiliation(s)
- Manas Sajjan
- Department of Chemistry, Purdue University, West Lafayette, IN-47907, USA. .,Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA
| | - Junxu Li
- Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA.,Department of Physics and Astronomy, Purdue University, West Lafayette, IN-47907, USA
| | - Raja Selvarajan
- Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA.,Department of Physics and Astronomy, Purdue University, West Lafayette, IN-47907, USA
| | - Shree Hari Sureshbabu
- Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA.,Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN-47907, USA
| | - Sumit Suresh Kale
- Department of Chemistry, Purdue University, West Lafayette, IN-47907, USA. .,Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA
| | - Rishabh Gupta
- Department of Chemistry, Purdue University, West Lafayette, IN-47907, USA. .,Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA
| | - Vinit Singh
- Department of Chemistry, Purdue University, West Lafayette, IN-47907, USA. .,Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA
| | - Sabre Kais
- Department of Chemistry, Purdue University, West Lafayette, IN-47907, USA. .,Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA.,Department of Physics and Astronomy, Purdue University, West Lafayette, IN-47907, USA.,Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN-47907, USA
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3
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Arceci L, Silvi P, Montangero S. Entanglement of Formation of Mixed Many-Body Quantum States via Tree Tensor Operators. PHYSICAL REVIEW LETTERS 2022; 128:040501. [PMID: 35148155 DOI: 10.1103/physrevlett.128.040501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/17/2020] [Accepted: 12/15/2021] [Indexed: 06/14/2023]
Abstract
We present a numerical strategy to efficiently estimate bipartite entanglement measures, and in particular the entanglement of formation, for many-body quantum systems on a lattice. Our approach exploits the tree tensor operator tensor network Ansatz, a positive loopless representation for density matrices which, as we demonstrate, efficiently encodes information on bipartite entanglement, enabling the upscaling of entanglement estimation. Employing this technique, we observe a finite-size scaling law for the entanglement of formation in 1D critical lattice models at finite temperature for up to 128 spins, extending to mixed states the scaling law for the entanglement entropy.
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Affiliation(s)
- L Arceci
- Dipartimento di Fisica e Astronomia "G. Galilei," Università di Padova, I-35131 Padova, Italy
- INFN, Sezione di Padova, I-35131 Padova, Italy
| | - P Silvi
- Center for Quantum Physics, Faculty of Mathematics, Computer Science and Physics, University of Innsbruck, A-6020 Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
| | - S Montangero
- Dipartimento di Fisica e Astronomia "G. Galilei," Università di Padova, I-35131 Padova, Italy
- INFN, Sezione di Padova, I-35131 Padova, Italy
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Jiménez JL, Crone SPG, Fogh E, Zayed ME, Lortz R, Pomjakushina E, Conder K, Läuchli AM, Weber L, Wessel S, Honecker A, Normand B, Rüegg C, Corboz P, Rønnow HM, Mila F. A quantum magnetic analogue to the critical point of water. Nature 2021; 592:370-375. [PMID: 33854247 DOI: 10.1038/s41586-021-03411-8] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2020] [Accepted: 02/26/2021] [Indexed: 02/02/2023]
Abstract
At the liquid-gas phase transition in water, the density has a discontinuity at atmospheric pressure; however, the line of these first-order transitions defined by increasing the applied pressure terminates at the critical point1, a concept ubiquitous in statistical thermodynamics2. In correlated quantum materials, it was predicted3 and then confirmed experimentally4,5 that a critical point terminates the line of Mott metal-insulator transitions, which are also first-order with a discontinuous charge carrier density. In quantum spin systems, continuous quantum phase transitions6 have been controlled by pressure7,8, applied magnetic field9,10 and disorder11, but discontinuous quantum phase transitions have received less attention. The geometrically frustrated quantum antiferromagnet SrCu2(BO3)2 constitutes a near-exact realization of the paradigmatic Shastry-Sutherland model12-14 and displays exotic phenomena including magnetization plateaus15, low-lying bound-state excitations16, anomalous thermodynamics17 and discontinuous quantum phase transitions18,19. Here we control both the pressure and the magnetic field applied to SrCu2(BO3)2 to provide evidence of critical-point physics in a pure spin system. We use high-precision specific-heat measurements to demonstrate that, as in water, the pressure-temperature phase diagram has a first-order transition line that separates phases with different local magnetic energy densities, and that terminates at an Ising critical point. We provide a quantitative explanation of our data using recently developed finite-temperature tensor-network methods17,20-22. These results further our understanding of first-order quantum phase transitions in quantum magnetism, with potential applications in materials where anisotropic spin interactions produce the topological properties23,24 that are useful for spintronic applications.
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Affiliation(s)
- J Larrea Jiménez
- Laboratory for Quantum Matter under Extreme Conditions, Institute of Physics, University of São Paulo, São Paulo, Brazil.,Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
| | - S P G Crone
- Institute for Theoretical Physics, University of Amsterdam, Amsterdam, The Netherlands.,Delta Institute for Theoretical Physics, University of Amsterdam, Amsterdam, The Netherlands
| | - E Fogh
- Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
| | - M E Zayed
- Department of Physics, Carnegie Mellon University in Qatar, Doha, Qatar
| | - R Lortz
- Department of Physics, Hong Kong University of Science and Technology, Kowloon, Hong Kong
| | - E Pomjakushina
- Laboratory for Multiscale Materials Experiments, Paul Scherrer Institute, Villigen-PSI, Switzerland
| | - K Conder
- Laboratory for Multiscale Materials Experiments, Paul Scherrer Institute, Villigen-PSI, Switzerland
| | - A M Läuchli
- Institut für Theoretische Physik, Universität Innsbruck, Innsbruck, Austria
| | - L Weber
- Institut für Theoretische Festkörperphysik, RWTH Aachen University, Aachen, Germany
| | - S Wessel
- Institut für Theoretische Festkörperphysik, RWTH Aachen University, Aachen, Germany
| | - A Honecker
- Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, CY Cergy Paris Université, Cergy-Pontoise, France
| | - B Normand
- Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland.,Paul Scherrer Institute, Villigen-PSI, Switzerland
| | - Ch Rüegg
- Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland.,Paul Scherrer Institute, Villigen-PSI, Switzerland.,Institute for Quantum Electronics, ETH Zürich, Hönggerberg, Switzerland.,Department of Quantum Matter Physics, University of Geneva, Geneva, Switzerland
| | - P Corboz
- Institute for Theoretical Physics, University of Amsterdam, Amsterdam, The Netherlands.,Delta Institute for Theoretical Physics, University of Amsterdam, Amsterdam, The Netherlands
| | - H M Rønnow
- Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland.
| | - F Mila
- Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
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Jahromi SS, Orús R. Thermal bosons in 3d optical lattices via tensor networks. Sci Rep 2020; 10:19051. [PMID: 33149156 PMCID: PMC7642398 DOI: 10.1038/s41598-020-75548-x] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/14/2020] [Accepted: 10/13/2020] [Indexed: 11/26/2022] Open
Abstract
Ultracold atoms in optical lattices are one of the most promising experimental setups to simulate strongly correlated systems. However, efficient numerical algorithms able to benchmark experiments at low-temperatures in interesting 3d lattices are lacking. To this aim, here we introduce an efficient tensor network algorithm to accurately simulate thermal states of local Hamiltonians in any infinite lattice, and in any dimension. We apply the method to simulate thermal bosons in optical lattices. In particular, we study the physics of the (soft-core and hard-core) Bose–Hubbard model on the infinite pyrochlore and cubic lattices with unprecedented accuracy. Our technique is therefore an ideal tool to benchmark realistic and interesting optical-lattice experiments.
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Affiliation(s)
- Saeed S Jahromi
- Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018, San Sebastián, Spain
| | - Román Orús
- Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018, San Sebastián, Spain. .,Ikerbasque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Spain. .,Multiverse Computing, Paseo de Miramón 170, 20014, San Sebastián, Spain.
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Shi T, Demler E, Cirac JI. Variational Approach for Many-Body Systems at Finite Temperature. PHYSICAL REVIEW LETTERS 2020; 125:180602. [PMID: 33196237 DOI: 10.1103/physrevlett.125.180602] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2020] [Revised: 07/02/2020] [Accepted: 09/18/2020] [Indexed: 06/11/2023]
Abstract
We introduce an equation for density matrices that ensures a monotonic decrease of the free energy and reaches a fixed point at the Gibbs thermal. We build a variational approach for many-body systems that can be applied to a broad class of states, including all bosonic and fermionic Gaussian, as well as their generalizations obtained by unitary transformations, such as polaron transformations in electron-phonon systems. We apply it to the Holstein model on 20×20 and 50×50 square lattices, and predict phase separation between the superconducting and charge-density wave phases in the strong interaction regime.
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Affiliation(s)
- Tao Shi
- Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Eugene Demler
- Department of Physics, Harvard University, 17 Oxford Street, Cambridge, Massachusetts 02138, USA
| | - J Ignacio Cirac
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse. 1, 85748 Garching, Germany
- Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 München, Germany
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Tang W, Tu HH, Wang L. Continuous Matrix Product Operator Approach to Finite Temperature Quantum States. PHYSICAL REVIEW LETTERS 2020; 125:170604. [PMID: 33156680 DOI: 10.1103/physrevlett.125.170604] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/08/2020] [Accepted: 09/16/2020] [Indexed: 06/11/2023]
Abstract
We present an algorithm for studying quantum systems at finite temperature using continuous matrix product operator representation. The approach handles both short-range and long-range interactions in the thermodynamic limit without incurring any time discretization error. Moreover, the approach provides direct access to physical observables including the specific heat, local susceptibility, and local spectral functions. After verifying the method using the prototypical quantum XXZ chains, we apply it to quantum Ising models with power-law decaying interactions and on the infinite cylinder, respectively. The approach offers predictions that are relevant to experiments in quantum simulators and the nuclear magnetic resonance spin-lattice relaxation rate.
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Affiliation(s)
- Wei Tang
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
| | - Hong-Hao Tu
- Institute of Theoretical Physics, Technische Universität Dresden, 01062 Dresden, Germany
| | - Lei Wang
- Beijing National Lab for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
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8
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Schmoll P, Jahromi SS, Hörmann M, Mühlhauser M, Schmidt KP, Orús R. Fine Grained Tensor Network Methods. PHYSICAL REVIEW LETTERS 2020; 124:200603. [PMID: 32501041 DOI: 10.1103/physrevlett.124.200603] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/19/2019] [Revised: 03/21/2020] [Accepted: 04/16/2020] [Indexed: 06/11/2023]
Abstract
We develop a strategy for tensor network algorithms that allows to deal very efficiently with lattices of high connectivity. The basic idea is to fine grain the physical degrees of freedom, i.e., decompose them into more fundamental units which, after a suitable coarse graining, provide the original ones. Thanks to this procedure, the original lattice with high connectivity is transformed by an isometry into a simpler structure, which is easier to simulate via usual tensor network methods. In particular this enables the use of standard schemes to contract infinite 2D tensor networks-such as corner transfer matrix renormalization schemes-which are more involved on complex lattice structures. We prove the validity of our approach by numerically computing the ground-state properties of the ferromagnetic spin-1 transverse-field Ising model on the 2D triangular and 3D stacked triangular lattice, as well as of the hardcore and softcore Bose-Hubbard models on the triangular lattice. Our results are benchmarked against those obtained with other techniques, such as perturbative continuous unitary transformations and graph projected entangled pair states, showing excellent agreement and also improved performance in several regimes.
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Affiliation(s)
- Philipp Schmoll
- Institute of Physics, Johannes Gutenberg University, 55099 Mainz, Germany
| | - Saeed S Jahromi
- Donostia International Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebastián, Spain
| | - Max Hörmann
- Chair for Theoretical Physics I, FAU Erlangen-Nürnberg, 91058 Erlangen, Germany
| | - Matthias Mühlhauser
- Chair for Theoretical Physics I, FAU Erlangen-Nürnberg, 91058 Erlangen, Germany
| | - Kai Phillip Schmidt
- Chair for Theoretical Physics I, FAU Erlangen-Nürnberg, 91058 Erlangen, Germany
| | - Román Orús
- Donostia International Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebastián, Spain
- Ikerbasque Foundation for Science, Maria Diaz de Haro 3, E-48013 Bilbao, Spain
- Multiverse Computing, Pio Baroja 37, 20008 San Sebastián, Spain
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