Search Menu
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information
Search Scholarly Publications

Title: On the exact continuous mapping of fermions

Abstract

We derive a rigorous, quantum mechanical map of fermionic creation and annihilation operators to continuous Cartesian variables that exactly reproduces the matrix structure of the many-fermion problem. We show how our scheme can be used to map a general many-fermion Hamiltonian and then consider two specific models that encode the fundamental physics of many fermionic systems, the Anderson impurity and Hubbard models. We use these models to demonstrate how efficient mappings of these Hamiltonians can be constructed using a judicious choice of index ordering of the fermions. This development provides an alternative exact route to calculate the static and dynamical properties of fermionic systems and sets the stage to exploit the quantum-classical and semiclassical hierarchies to systematically derive methods offering a range of accuracies, thus enabling the study of problems where the fermionic degrees of freedom are coupled to complex anharmonic nuclear motion and spins which lie beyond the reach of most currently available methods.

Authors:
 [1]; ORCiD logo [1]
+ Show Author Affiliations
  1. Stanford Univ., CA (United States). Dept. of Chemistry
Publication Date:
Research Org.:
Stanford Univ., CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES). Chemical Sciences, Geosciences, and Biosciences Division
OSTI Identifier:
1499991
Alternate Identifier(s):
OSTI ID: 1526793
Grant/Contract Number:  
SC0014437
Resource Type:
Accepted Manuscript
Journal Name:
Scientific Reports
Additional Journal Information:
Journal Volume: 8; Journal ID: ISSN 2045-2322
Publisher:
Nature Publishing Group
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; chemical physics; method development; quantum information; theoretical physics

Citation Formats

Montoya-Castillo, Andrés, and Markland, Thomas E. On the exact continuous mapping of fermions. United States: N. p., 2018. Web. doi:10.1038/s41598-018-31162-6.
Copy to clipboard
Montoya-Castillo, Andrés, & Markland, Thomas E. On the exact continuous mapping of fermions. United States. https://doi.org/10.1038/s41598-018-31162-6
Copy to clipboard
Montoya-Castillo, Andrés, and Markland, Thomas E. Tue . "On the exact continuous mapping of fermions". United States. https://doi.org/10.1038/s41598-018-31162-6. https://www.osti.gov/servlets/purl/1499991.
Copy to clipboard
@article{osti_1499991,
title = {On the exact continuous mapping of fermions},
author = {Montoya-Castillo, Andrés and Markland, Thomas E.},
abstractNote = {We derive a rigorous, quantum mechanical map of fermionic creation and annihilation operators to continuous Cartesian variables that exactly reproduces the matrix structure of the many-fermion problem. We show how our scheme can be used to map a general many-fermion Hamiltonian and then consider two specific models that encode the fundamental physics of many fermionic systems, the Anderson impurity and Hubbard models. We use these models to demonstrate how efficient mappings of these Hamiltonians can be constructed using a judicious choice of index ordering of the fermions. This development provides an alternative exact route to calculate the static and dynamical properties of fermionic systems and sets the stage to exploit the quantum-classical and semiclassical hierarchies to systematically derive methods offering a range of accuracies, thus enabling the study of problems where the fermionic degrees of freedom are coupled to complex anharmonic nuclear motion and spins which lie beyond the reach of most currently available methods.},
doi = {10.1038/s41598-018-31162-6},
journal = {Scientific Reports},
number = ,
volume = 8,
place = {United States},
year = {Tue Aug 28 00:00:00 EDT 2018},
month = {Tue Aug 28 00:00:00 EDT 2018}
}
Copy to clipboard
Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1038/s41598-018-31162-6
Copyright Statement
Other availability
Search WorldCat Search WorldCat to find libraries that may hold this journal
Citation Metrics:
Cited by: 3 works
Citation information provided by
Web of Science

Figures / Tables:

Figure 1 Figure 1: Index ordering options for the Anderson model.
All figures and tables (2 total)
Save / Share:
Export Metadata
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.

Works referenced in this record:

Iterative linearized approach to nonadiabatic dynamics
journal, September 2008

  • Dunkel, E. R.; Bonella, S.; Coker, D. F.
  • The Journal of Chemical Physics, Vol. 129, Issue 11
  • DOI: 10.1063/1.2976441

A classical analog for electronic degrees of freedom in nonadiabatic collision processes
journal, April 1979

  • Meyera), Hans‐Dieter; Miller, William H.
  • The Journal of Chemical Physics, Vol. 70, Issue 7
  • DOI: 10.1063/1.437910

Mixed semiclassical–classical approaches to the dynamics of complex molecular systems
journal, January 1997

  • Sun, Xiong; Miller, William H.
  • The Journal of Chemical Physics, Vol. 106, Issue 3
  • DOI: 10.1063/1.473171

Digital quantum simulation of fermionic models with a superconducting circuit
journal, July 2015

  • Barends, R.; Lamata, L.; Kelly, J.
  • Nature Communications, Vol. 6, Issue 1
  • DOI: 10.1038/ncomms8654

Gaussian quantum information
journal, May 2012

  • Weedbrook, Christian; Pirandola, Stefano; García-Patrón, Raúl
  • Reviews of Modern Physics, Vol. 84, Issue 2
  • DOI: 10.1103/RevModPhys.84.621

The semiclassical propagator in fermionic Fock space
journal, September 2014

  • Engl, Thomas; Plößl, Peter; Urbina, Juan Diego
  • Theoretical Chemistry Accounts, Vol. 133, Issue 11
  • DOI: 10.1007/s00214-014-1563-9

A Cartesian classical second-quantized many-electron Hamiltonian, for use with the semiclassical initial value representation
journal, October 2012

  • Li, Bin; Miller, William H.
  • The Journal of Chemical Physics, Vol. 137, Issue 15
  • DOI: 10.1063/1.4757935

Communication: Partial linearized density matrix dynamics for dissipative, non-adiabatic quantum evolution
journal, November 2011

  • Huo, Pengfei; Coker, David F.
  • The Journal of Chemical Physics, Vol. 135, Issue 20
  • DOI: 10.1063/1.3664763

Mapping variable ring polymer molecular dynamics: A path-integral based method for nonadiabatic processes
journal, September 2013

  • Ananth, Nandini
  • The Journal of Chemical Physics, Vol. 139, Issue 12
  • DOI: 10.1063/1.4821590

Fröhlich polaron and bipolaron: recent developments
journal, May 2009

Studies of polaron motion
journal, November 1959

A Lattice Model of Liquid Helium, III: Equilibrium Properties of Liquid He
journal, October 1957

  • Matsuda, Hirotsugu
  • Progress of Theoretical Physics, Vol. 18, Issue 4
  • DOI: 10.1143/PTP.18.357

Classical models for electronic degrees of freedom: The second‐quantized many‐electron Hamiltonian
journal, May 1986

  • Miller, William H.; White, Kim A.
  • The Journal of Chemical Physics, Vol. 84, Issue 9
  • DOI: 10.1063/1.450655

Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet
journal, December 1940

Molecular dynamics for fermions
journal, July 2000

Mapping quantum-classical Liouville equation: Projectors and trajectories
journal, February 2012

  • Kelly, Aaron; van Zon, Ramses; Schofield, Jeremy
  • The Journal of Chemical Physics, Vol. 136, Issue 8
  • DOI: 10.1063/1.3685420

LAND-map, a linearized approach to nonadiabatic dynamics using the mapping formalism
journal, May 2005

  • Bonella, S.; Coker, D. F.
  • The Journal of Chemical Physics, Vol. 122, Issue 19
  • DOI: 10.1063/1.1896948

Semiclassical approximations for the calculation of thermal rate constants for chemical reactions in complex molecular systems
journal, June 1998

  • Wang, Haobin; Sun, Xiong; Miller, William H.
  • The Journal of Chemical Physics, Vol. 108, Issue 23
  • DOI: 10.1063/1.476447

Bipolarons
journal, December 1994

Effect of Correlation on the Ferromagnetism of Transition Metals
journal, May 1964

Gaussian operator bases for correlated fermions
journal, December 2005

Gaussian phase-space representations for fermions
journal, March 2006

Studies of Polaron Motion
journal, April 2000

Quantum simulation
journal, March 2014

Exact quantum statistics for electronically nonadiabatic systems using continuous path variables
journal, December 2010

  • Ananth, Nandini; Miller, Thomas F.
  • The Journal of Chemical Physics, Vol. 133, Issue 23
  • DOI: 10.1063/1.3511700

Communication: Nonadiabatic ring-polymer molecular dynamics
journal, July 2013

  • Richardson, Jeremy O.; Thoss, Michael
  • The Journal of Chemical Physics, Vol. 139, Issue 3
  • DOI: 10.1063/1.4816124

Quantum-classical Liouville dynamics in the mapping basis
journal, August 2008

  • Kim, Hyojoon; Nassimi, Ali; Kapral, Raymond
  • The Journal of Chemical Physics, Vol. 129, Issue 8
  • DOI: 10.1063/1.2971041

Semiclassical description of nonadiabatic quantum dynamics: Application to the S1–S2 conical intersection in pyrazine
journal, June 2000

  • Thoss, Michael; Miller, William H.; Stock, Gerhard
  • The Journal of Chemical Physics, Vol. 112, Issue 23
  • DOI: 10.1063/1.481668

Exponentially more precise quantum simulation of fermions in second quantization
journal, March 2016

Nonadiabatic dynamics in open quantum-classical systems: Forward-backward trajectory solution
journal, December 2012

  • Hsieh, Chang-Yu; Kapral, Raymond
  • The Journal of Chemical Physics, Vol. 137, Issue 22
  • DOI: 10.1063/1.4736841

A Centroid Molecular Dynamics Approach for Nonadiabatic Dynamical Processes in Condensed Phases:  the Spin-Boson Case †
journal, August 2002

  • Liao, Jie-Lou; Voth, Gregory A.
  • The Journal of Physical Chemistry B, Vol. 106, Issue 33
  • DOI: 10.1021/jp020978d

A semiclassical generalized quantum master equation for an arbitrary system-bath coupling
journal, June 2004

  • Shi, Qiang; Geva, Eitan
  • The Journal of Chemical Physics, Vol. 120, Issue 22
  • DOI: 10.1063/1.1738109

Semiclassical description of electronically nonadiabatic dynamics via the initial value representation
journal, August 2007

  • Ananth, Nandini; Venkataraman, Charulatha; Miller, William H.
  • The Journal of Chemical Physics, Vol. 127, Issue 8
  • DOI: 10.1063/1.2759932

Many-Body Effects on the Transport Properties of Single-Molecule Devices
journal, September 2004

Consistent treatment of quantum-mechanical and classical degrees of freedom in mixed quantum-classical simulations
journal, May 1998

  • Müller, Uwe; Stock, Gerhard
  • The Journal of Chemical Physics, Vol. 108, Issue 18
  • DOI: 10.1063/1.476184

Coherent State Quantization of Constraint Systems
journal, March 1997

Deriving the exact nonadiabatic quantum propagator in the mapping variable representation
journal, January 2016

  • Hele, Timothy J. H.; Ananth, Nandini
  • Faraday Discussions, Vol. 195
  • DOI: 10.1039/C6FD00106H

Gaussian Quantum Monte Carlo Methods for Fermions and Bosons
journal, December 2004

Phonon Spectrum Changes in Small Particles and Their Implications for Superconductivity
journal, November 1968

Quantum algorithms for fermionic simulations
journal, July 2001

Classical mapping for Hubbard operators: Application to the double-Anderson model
journal, May 2014

  • Li, Bin; Miller, William H.; Levy, Tal J.
  • The Journal of Chemical Physics, Vol. 140, Issue 20
  • DOI: 10.1063/1.4878736

Nonadiabatic semiclassical dynamics in the mixed quantum-classical initial value representation
journal, March 2018

  • Church, Matthew S.; Hele, Timothy J. H.; Ezra, Gregory S.
  • The Journal of Chemical Physics, Vol. 148, Issue 10
  • DOI: 10.1063/1.5005557

Efficient Dynamical Mean Field Simulation of the Holstein-Hubbard Model
journal, October 2007

Continuous surface switching: An improved time-dependent self-consistent-field method for nonadiabatic dynamics
journal, June 2000

  • Volobuev, Yuri L.; Hack, Michael D.; Topaler, Maria S.
  • The Journal of Chemical Physics, Vol. 112, Issue 22
  • DOI: 10.1063/1.481609

Correlated electrons in high-temperature superconductors
journal, July 1994

Flow of zero-point energy and exploration of phase space in classical simulations of quantum relaxation dynamics
journal, July 1999

  • Stock, Gerhard; Müller, Uwe
  • The Journal of Chemical Physics, Vol. 111, Issue 1
  • DOI: 10.1063/1.479254

Mapping approach to the semiclassical description of nonadiabatic quantum dynamics
journal, January 1999

Electron Correlation and Ferromagnetism of Transition Metals
journal, September 1963

  • Kanamori, Junjiro
  • Progress of Theoretical Physics, Vol. 30, Issue 3
  • DOI: 10.1143/PTP.30.275

Coherent state mapping ring polymer molecular dynamics for non-adiabatic quantum propagations
journal, December 2017

  • Chowdhury, Sutirtha N.; Huo, Pengfei
  • The Journal of Chemical Physics, Vol. 147, Issue 21
  • DOI: 10.1063/1.4995616

Grassmann phase-space methods for fermions: Uncovering the classical probability structure
journal, December 2016

Boson realizations of Lie algebras with applications to nuclear physics
journal, April 1991

The method of Boson expansions in quantum theory
journal, February 1978

Grassmann phase space methods for fermions. I. Mode theory
journal, July 2016

Mott Transition in the Two-Dimensional Hubbard Model
journal, February 2012

Semiclassical approach to dynamics of interacting fermions
journal, September 2017

Electron correlations in narrow energy bands
journal, November 1963

  • Hubbard, J.
  • Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 276, Issue 1365, p. 238-257
  • DOI: 10.1098/rspa.1963.0204

Doping a Mott insulator: Physics of high-temperature superconductivity
journal, January 2006

  • Lee, Patrick A.; Nagaosa, Naoto; Wen, Xiao-Gang
  • Reviews of Modern Physics, Vol. 78, Issue 1
  • DOI: 10.1103/RevModPhys.78.17

Theory of Localized Magnetic States in Metals
journal, March 1966

  • Anderson, P. W.
  • Journal of Applied Physics, Vol. 37, Issue 3
  • DOI: 10.1063/1.1708389

Application of a semiclassical model for the second-quantized many-electron Hamiltonian to nonequilibrium quantum transport: The resonant level model
journal, April 2011

  • Swenson, David W. H.; Levy, Tal; Cohen, Guy
  • The Journal of Chemical Physics, Vol. 134, Issue 16
  • DOI: 10.1063/1.3583366

Simulating physical phenomena by quantum networks
journal, April 2002

Semiclassical theory of electronically nonadiabatic dynamics: Results of a linearized approximation to the initial value representation
journal, November 1998

  • Sun, Xiong; Wang, Haobin; Miller, William H.
  • The Journal of Chemical Physics, Vol. 109, Issue 17
  • DOI: 10.1063/1.477389

Nonequilibrium electron transport in strongly correlated molecular junctions
journal, March 2010

Effect of Correlation on the Ferromagnetism of Transition Metals
journal, March 1963

Localized Magnetic States in Metals
journal, October 1961

A Lattice Model of Liquid Helium, I
journal, December 1956

  • Matsubara, Takeo; Matsuda, Hirotsugu
  • Progress of Theoretical Physics, Vol. 16, Issue 6
  • DOI: 10.1143/PTP.16.569

Fragment formation studied with antisymmetrized version of molecular dynamics with two-nucleon collisions
journal, May 1992

Studies of polaron motion
journal, November 1959

Deriving the exact nonadiabatic quantum propagator in the mapping variable representation
text, January 2016

  • Hele, Tim; Ananth, N.
  • Apollo - University of Cambridge Repository
  • DOI: 10.17863/cam.11879

Nonadiabatic semiclassical dynamics in the mixed quantum-classical initial value representation.
text, January 2018

  • Church, Matthew S.; Hele, Tim; Ezra, Gregory S.
  • Apollo - University of Cambridge Repository
  • DOI: 10.17863/cam.17761

The semiclassical propagator in fermionic Fock space
text, January 2014

  • Engl, Thomas; Plößl, Peter; Urbina, Juan Diego
  • Universität Regensburg
  • DOI: 10.5283/epub.30753

Quantum Physics in One Dimension
journal, April 2004

Quantum-Classical Liouville Dynamics in the Mapping Basis
text, January 2009

Froehlich Polaron and Bipolaron: Recent Developments
text, January 2009

Mapping quantum-classical Liouville equation: projectors and trajectories
text, January 2012

The semiclassical propagator in fermionic Fock space
text, January 2014

Semiclassical approach to Dynamics of Interacting Fermions
text, January 2016

Nonadiabatic semiclassical dynamics in the mixed quantum-classical initial value representation
text, January 2017

Simulating Physical Phenomena by Quantum Networks
text, January 2001

Gaussian operator bases for correlated fermions
text, January 2005

Grassmann phase space methods for fermions. II. Field theory
journal, February 2017

A Centroid Molecular Dynamics Approach for Nonadiabatic Dynamical Processes in Condensed Phases:  the Spin-Boson Case †
journal, August 2002

  • Liao, Jie-Lou; Voth, Gregory A.
  • The Journal of Physical Chemistry B, Vol. 106, Issue 33
  • DOI: 10.1021/jp020978d

Digital quantum simulation of fermionic models with a superconducting circuit
journal, July 2015

  • Barends, R.; Lamata, L.; Kelly, J.
  • Nature Communications, Vol. 6, Issue 1
  • DOI: 10.1038/ncomms8654

A semiclassical generalized quantum master equation for an arbitrary system-bath coupling
journal, June 2004

  • Shi, Qiang; Geva, Eitan
  • The Journal of Chemical Physics, Vol. 120, Issue 22
  • DOI: 10.1063/1.1738109

LAND-map, a linearized approach to nonadiabatic dynamics using the mapping formalism
journal, May 2005

  • Bonella, S.; Coker, D. F.
  • The Journal of Chemical Physics, Vol. 122, Issue 19
  • DOI: 10.1063/1.1896948

Semiclassical description of electronically nonadiabatic dynamics via the initial value representation
journal, August 2007

  • Ananth, Nandini; Venkataraman, Charulatha; Miller, William H.
  • The Journal of Chemical Physics, Vol. 127, Issue 8
  • DOI: 10.1063/1.2759932

Quantum-classical Liouville dynamics in the mapping basis
journal, August 2008

  • Kim, Hyojoon; Nassimi, Ali; Kapral, Raymond
  • The Journal of Chemical Physics, Vol. 129, Issue 8
  • DOI: 10.1063/1.2971041

Mapping quantum-classical Liouville equation: Projectors and trajectories
journal, February 2012

  • Kelly, Aaron; van Zon, Ramses; Schofield, Jeremy
  • The Journal of Chemical Physics, Vol. 136, Issue 8
  • DOI: 10.1063/1.3685420

Mixed semiclassical–classical approaches to the dynamics of complex molecular systems
journal, January 1997

  • Sun, Xiong; Miller, William H.
  • The Journal of Chemical Physics, Vol. 106, Issue 3
  • DOI: 10.1063/1.473171

Nonadiabatic dynamics in open quantum-classical systems: Forward-backward trajectory solution
journal, December 2012

  • Hsieh, Chang-Yu; Kapral, Raymond
  • The Journal of Chemical Physics, Vol. 137, Issue 22
  • DOI: 10.1063/1.4736841

A Cartesian classical second-quantized many-electron Hamiltonian, for use with the semiclassical initial value representation
journal, October 2012

  • Li, Bin; Miller, William H.
  • The Journal of Chemical Physics, Vol. 137, Issue 15
  • DOI: 10.1063/1.4757935

Semiclassical approximations for the calculation of thermal rate constants for chemical reactions in complex molecular systems
journal, June 1998

  • Wang, Haobin; Sun, Xiong; Miller, William H.
  • The Journal of Chemical Physics, Vol. 108, Issue 23
  • DOI: 10.1063/1.476447

Flow of zero-point energy and exploration of phase space in classical simulations of quantum relaxation dynamics
journal, July 1999

  • Stock, Gerhard; Müller, Uwe
  • The Journal of Chemical Physics, Vol. 111, Issue 1
  • DOI: 10.1063/1.479254

Continuous surface switching: An improved time-dependent self-consistent-field method for nonadiabatic dynamics
journal, June 2000

  • Volobuev, Yuri L.; Hack, Michael D.; Topaler, Maria S.
  • The Journal of Chemical Physics, Vol. 112, Issue 22
  • DOI: 10.1063/1.481609

Semiclassical description of nonadiabatic quantum dynamics: Application to the S1–S2 conical intersection in pyrazine
journal, June 2000

  • Thoss, Michael; Miller, William H.; Stock, Gerhard
  • The Journal of Chemical Physics, Vol. 112, Issue 23
  • DOI: 10.1063/1.481668

Mapping variable ring polymer molecular dynamics: A path-integral based method for nonadiabatic processes
journal, September 2013

  • Ananth, Nandini
  • The Journal of Chemical Physics, Vol. 139, Issue 12
  • DOI: 10.1063/1.4821590

Classical mapping for Hubbard operators: Application to the double-Anderson model
journal, May 2014

  • Li, Bin; Miller, William H.; Levy, Tal J.
  • The Journal of Chemical Physics, Vol. 140, Issue 20
  • DOI: 10.1063/1.4878736

Nonadiabatic semiclassical dynamics in the mixed quantum-classical initial value representation
journal, March 2018

  • Church, Matthew S.; Hele, Timothy J. H.; Ezra, Gregory S.
  • The Journal of Chemical Physics, Vol. 148, Issue 10
  • DOI: 10.1063/1.5005557

Bipolarons
journal, December 1994

Langevin equations for interacting fermions and Cooper-like pairing in trapped one-dimensional fermions
journal, November 2001

Quantum dynamics with fermion coupled coherent states: Theory and application to electron dynamics in laser fields
journal, September 2011

Comment on “Gaussian Quantum Monte Carlo Methods for Fermions and Bosons”
journal, May 2006

A Lattice Model of Liquid Helium
journal, October 1956

  • Matsubara, Takeo; Matsuda, Hirotsugu
  • Progress of Theoretical Physics, Vol. 16, Issue 4
  • DOI: 10.1143/ptp.16.416

Froehlich Polaron and Bipolaron: Recent Developments
text, January 2009

The semiclassical propagator in fermionic Fock space
text, January 2014

Exponentially more precise quantum simulation of fermions I: Quantum chemistry in second quantization
text, January 2015

Semiclassical approach to Dynamics of Interacting Fermions
text, January 2016

Fermi softness: a local perspective on surface reactivity
preprint, January 2020

Many Body Effects on the Transport Properties of Single-Molecule Devices
text, January 2004

Gaussian phase-space representations for fermions
text, January 2004

Gaussian operator bases for correlated fermions
text, January 2005

    Sort by:
    [ × clear filter / sort ]

    Works referencing / citing this record:

    A complete quasiclassical map for the dynamics of interacting fermions
    journal, June 2019

    • Levy, Amikam; Dou, Wenjie; Rabani, Eran
    • The Journal of Chemical Physics, Vol. 150, Issue 23
    • DOI: 10.1063/1.5099987

    A complete quasiclassical map for the dynamics of interacting fermions
    text, January 2019

      Sort by:
      [ × clear filter / sort ]

      Figures / Tables found in this record:

      Figure 1 (p. 6)figure
      Figure 2 (p. 7)figure
        [ × clear filter / sort ]
        Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.

        Similar Records in DOE PAGES and OSTI.GOV collections: