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Martyn JM, Najafi K, Luo D. Variational Neural-Network Ansatz for Continuum Quantum Field Theory. PHYSICAL REVIEW LETTERS 2023; 131:081601. [PMID: 37683171 DOI: 10.1103/physrevlett.131.081601] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/16/2022] [Revised: 07/21/2023] [Accepted: 07/25/2023] [Indexed: 09/10/2023]
Abstract
Physicists dating back to Feynman have lamented the difficulties of applying the variational principle to quantum field theories. In nonrelativistic quantum field theories, the challenge is to parametrize and optimize over the infinitely many n-particle wave functions comprising the state's Fock-space representation. Here we approach this problem by introducing neural-network quantum field states, a deep learning ansatz that enables application of the variational principle to nonrelativistic quantum field theories in the continuum. Our ansatz uses the Deep Sets neural network architecture to simultaneously parametrize all of the n-particle wave functions comprising a quantum field state. We employ our ansatz to approximate ground states of various field theories, including an inhomogeneous system and a system with long-range interactions, thus demonstrating a powerful new tool for probing quantum field theories.
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Affiliation(s)
- John M Martyn
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
- The NSF AI Institute for Artificial Intelligence and Fundamental Interactions
| | - Khadijeh Najafi
- IBM Quantum, IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA
- MIT-IBM Watson AI Lab, Cambridge, Massachusetts 02142, USA
| | - Di Luo
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
- The NSF AI Institute for Artificial Intelligence and Fundamental Interactions
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
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2
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Dutta S, Buyskikh A, Daley AJ, Mueller EJ. Density Matrix Renormalization Group for Continuous Quantum Systems. PHYSICAL REVIEW LETTERS 2022; 128:230401. [PMID: 35749177 DOI: 10.1103/physrevlett.128.230401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/20/2021] [Revised: 03/22/2022] [Accepted: 05/06/2022] [Indexed: 06/15/2023]
Abstract
We introduce a versatile and practical framework for applying matrix product state techniques to continuous quantum systems. We divide space into multiple segments and generate continuous basis functions for the many-body state in each segment. By combining this mapping with existing numerical density matrix renormalization group routines, we show how one can accurately obtain the ground-state wave function, spatial correlations, and spatial entanglement entropy directly in the continuum. For a prototypical mesoscopic system of strongly interacting bosons we demonstrate faster convergence than standard grid-based discretization. We illustrate the power of our approach by studying a superfluid-insulator transition in an external potential. We outline how one can directly apply or generalize this technique to a wide variety of experimentally relevant problems across condensed matter physics and quantum field theory.
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Affiliation(s)
- Shovan Dutta
- T.C.M. Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA
- Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
| | - Anton Buyskikh
- Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
| | - Andrew J Daley
- Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
| | - Erich J Mueller
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA
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Beau M, Pittman SM, Astrakharchik GE, Del Campo A. Exactly Solvable System of One-Dimensional Trapped Bosons with Short- and Long-Range Interactions. PHYSICAL REVIEW LETTERS 2020; 125:220602. [PMID: 33315444 DOI: 10.1103/physrevlett.125.220602] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2020] [Accepted: 10/30/2020] [Indexed: 06/12/2023]
Abstract
We introduce a model of trapped bosons with contact interactions as well as Coulomb repulsion or gravitational attraction in one spatial dimension. We find the exact ground-state energy and many-body wave function. The density profile and the pair-correlation function are sampled using Monte Carlo method and show a rich variety of regimes with crossovers between them. Strong attraction leads to a trapped McGuire quantum soliton. Weak repulsion results in an incompressible Laughlin-like fluid with flat density, well reproduced by a Gross-Pitaevskii equation with long-range interactions. Stronger repulsion induces Friedel oscillations and the eventual formation of a Wigner crystal.
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Affiliation(s)
- M Beau
- Department of Physics, University of Massachusetts, Boston, Massachusetts 02125, USA
- Dublin Institute for Advanced Studies, School of Theoretical Physics, 10 Burlington Road, Dublin 4, Ireland
| | - S M Pittman
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - G E Astrakharchik
- Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain
| | - A Del Campo
- Department of Physics, University of Massachusetts, Boston, Massachusetts 02125, USA
- Donostia International Physics Center, E-20018 San Sebastián, Spain
- IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao, Spain
- Theory Division, Los Alamos National Laboratory, MS-B213, Los Alamos, New Mexico 87545, USA
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Liu Y, Qi F, Zhang Y, Chen S. Mass-Imbalanced Atoms in a Hard-Wall Trap: An Exactly Solvable Model Associated with D 6 Symmetry. iScience 2019; 22:181-194. [PMID: 31785556 PMCID: PMC6911986 DOI: 10.1016/j.isci.2019.11.018] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2019] [Revised: 08/16/2019] [Accepted: 11/09/2019] [Indexed: 11/13/2022] Open
Abstract
We show that a system consisting of two interacting particles with mass ratio 3 or 1/3 in a hard-wall box can be exactly solved by using Bethe-type ansatz. The ansatz is based on a finite superposition of plane waves associated with a dihedral group D6, which enforces the momentums after a series of scattering and reflection processes to fulfill the D6 symmetry. Starting from a two-body elastic collision model in a hard-wall box, we demonstrate how a finite momentum distribution is related to the D2n symmetry for permitted mass ratios. For a quantum system with mass ratio 3, we obtain exact eigenenergies and eigenstates by solving Bethe-type-ansatz equations for arbitrary interaction strength. A many-body excited state of the system is found to be independent of the interaction strength, i.e., the wave function looks exactly the same for non-interacting two particles or in the hard-core limit. At nonergodicity points, the permitted momentums fulfill the D2n symmetry The Bethe Ansatz method is extended to solve the mass-imbalanced quantum system We exactly solve the interacting two-particle system in a hard-wall with mass ratio 3
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Affiliation(s)
- Yanxia Liu
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China; Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, P. R. China
| | - Fan Qi
- Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, P. R. China
| | - Yunbo Zhang
- Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, P. R. China.
| | - Shu Chen
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China; School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China; The Yangtze River Delta Physics Research Center, Liyang, Jiangsu 213300, China.
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Reichert B, Astrakharchik GE, Petković A, Ristivojevic Z. Exact Results for the Boundary Energy of One-Dimensional Bosons. PHYSICAL REVIEW LETTERS 2019; 123:250602. [PMID: 31922773 DOI: 10.1103/physrevlett.123.250602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/21/2019] [Indexed: 06/10/2023]
Abstract
We study bosons in a one-dimensional hard-wall box potential. In the case of contact interaction, the system is exactly solvable by the Bethe ansatz, as first shown by Gaudin in 1971. Although contained in the exact solution, the boundary energy in the thermodynamic limit for this problem is only approximately calculated by Gaudin, who found the leading order result at weak repulsion. Here we derive an exact integral equation that enables one to calculate the boundary energy in the thermodynamic limit at an arbitrary interaction. We then solve such an equation and find the asymptotic results for the boundary energy at weak and strong interactions. The analytical results obtained from the Bethe ansatz are in agreement with the ones found by other complementary methods, including quantum Monte Carlo simulations. We study the universality of the boundary energy in the regime of a small gas parameter by making a comparison with the exact solution for the hard rod gas.
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Affiliation(s)
- Benjamin Reichert
- Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, 31062 Toulouse, France
| | - Grigori E Astrakharchik
- Departamento de Física, Universitat Politèecnica de Catalunya, Campus Nord B4-B5, 08034 Barcelona, Spain
| | - Aleksandra Petković
- Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, 31062 Toulouse, France
| | - Zoran Ristivojevic
- Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, 31062 Toulouse, France
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Sowiński T, Ángel García-March M. One-dimensional mixtures of several ultracold atoms: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2019; 82:104401. [PMID: 31404916 DOI: 10.1088/1361-6633/ab3a80] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Recent theoretical and experimental progress on studying one-dimensional systems of bosonic, fermionic, and Bose-Fermi mixtures of a few ultracold atoms confined in traps is reviewed in the broad context of mesoscopic quantum physics. We pay special attention to limiting cases of very strong or very weak interactions and transitions between them. For bosonic mixtures, we describe the developments in systems of three and four atoms as well as different extensions to larger numbers of particles. We also briefly review progress in the case of spinor Bose gases of a few atoms. For fermionic mixtures, we discuss a special role of spin and present a detailed discussion of the two- and three-atom cases. We discuss the advantages and disadvantages of different computation methods applied to systems with intermediate interactions. In the case of very strong repulsion, close to the infinite limit, we discuss approaches based on effective spin chain descriptions. We also report on recent studies on higher-spin mixtures and inter-component attractive forces. For both statistics, we pay particular attention to impurity problems and mass imbalance cases. Finally, we describe the recent advances on trapped Bose-Fermi mixtures, which allow for a theoretical combination of previous concepts, well illustrating the importance of quantum statistics and inter-particle interactions. Lastly, we report on fundamental questions related to the subject which we believe will inspire further theoretical developments and experimental verification.
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Affiliation(s)
- Tomasz Sowiński
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, PL-02668 Warsaw, Poland
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Geiger B, Urbina JD, Hummel Q, Richter K. Semiclassics in a system without classical limit: The few-body spectrum of two interacting bosons in one dimension. Phys Rev E 2017; 96:022204. [PMID: 28950541 DOI: 10.1103/physreve.96.022204] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/29/2017] [Indexed: 11/07/2022]
Abstract
We present a semiclassical study of the spectrum of a few-body system consisting of two short-range interacting bosonic particles in one dimension, a particular case of a general class of integrable many-body systems where the energy spectrum is given by the solution of algebraic transcendental equations. By an exact mapping between δ-potentials and boundary conditions on the few-body wave functions, we are able to extend previous semiclassical results for single-particle systems with mixed boundary conditions to the two-body problem. The semiclassical approach allows us to derive explicit analytical results for the smooth part of the two-body density of states that are in excellent agreement with numerical calculations. It further enables us to include the effect of bound states in the attractive case. Remarkably, for the particular case of two particles in one dimension, the discrete energy levels obtained through a requantization condition of the smooth density of states are essentially in perfect agreement with the exact ones.
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Affiliation(s)
- Benjamin Geiger
- Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
| | - Juan-Diego Urbina
- Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
| | - Quirin Hummel
- Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
| | - Klaus Richter
- Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
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Tokuno A, Oshikawa M, Demler E. Dynamics of one-dimensional Bose liquids: Andreev-like reflection at Y junctions and the absence of the Aharonov-Bohm effect. PHYSICAL REVIEW LETTERS 2008; 100:140402. [PMID: 18518007 DOI: 10.1103/physrevlett.100.140402] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2007] [Indexed: 05/26/2023]
Abstract
We study one-dimensional Bose liquids of interacting ultracold atoms in the Y-shaped potential when each branch is filled with atoms. We find that the excitation packet incident on a single Y junction should experience a negative density reflection analogous to the Andreev reflection at normal-superconductor interfaces, although the present system does not contain fermions. In a ring-interferometer-type configuration, we find that the transport is completely insensitive to the (effective) flux contained in the ring, in contrast with the Aharonov-Bohm effect of a single particle in the same geometry.
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Affiliation(s)
- Akiyuki Tokuno
- Institute for Solid State Physics, University of Tokyo, Kashiwa 227-8581 Japan and Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551 Japan
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Buljan H, Pezer R, Gasenzer T. Fermi-bose transformation for the time-dependent Lieb-Liniger gas. PHYSICAL REVIEW LETTERS 2008; 100:080406. [PMID: 18352608 DOI: 10.1103/physrevlett.100.080406] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/10/2007] [Revised: 12/05/2007] [Indexed: 05/26/2023]
Abstract
Exact solutions of the Schrödinger equation describing a freely expanding Lieb-Liniger gas of delta-interacting bosons in one spatial dimension are constructed. We demonstrate that for any interaction strength the system enters a strongly correlated regime during such expansion. The asymptotic form of the wave function is shown to have the form characteristic for "impenetrable-core" bosons. Exact solutions are obtained by transforming a fully antisymmetric (fermionic) time-dependent wave function that obeys the Schrödinger equation for a free gas. This transformation employs a differential Fermi-Bose mapping operator that depends on the strength of the interaction and the number of particles.
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Affiliation(s)
- H Buljan
- Department of Physics, University of Zagreb, PP 332, 10000 Zagreb, Croatia.
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Oelkers N, Batchelor MT, Bortz M, Guan XW. Bethe ansatz study of one-dimensional Bose and Fermi gases with periodic and hard wall boundary conditions. ACTA ACUST UNITED AC 2006. [DOI: 10.1088/0305-4470/39/5/005] [Citation(s) in RCA: 65] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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