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Erpenbeck A, Gull E, Cohen G. Quantum Monte Carlo Method in the Steady State. PHYSICAL REVIEW LETTERS 2023; 130:186301. [PMID: 37204908 DOI: 10.1103/physrevlett.130.186301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2022] [Revised: 12/07/2022] [Accepted: 04/07/2023] [Indexed: 05/21/2023]
Abstract
We present a numerically exact steady-state inchworm Monte Carlo method for nonequilibrium quantum impurity models. Rather than propagating an initial state to long times, the method is directly formulated in the steady state. This eliminates any need to traverse the transient dynamics and grants access to a much larger range of parameter regimes at vastly reduced computational costs. We benchmark the method on equilibrium Green's functions of quantum dots in the noninteracting limit and in the unitary limit of the Kondo regime. We then consider correlated materials described with dynamical mean field theory and driven away from equilibrium by a bias voltage. We show that the response of a correlated material to a bias voltage differs qualitatively from the splitting of the Kondo resonance observed in bias-driven quantum dots.
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Affiliation(s)
- A Erpenbeck
- Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
| | - E Gull
- Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
| | - G Cohen
- The Raymond and Beverley Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel
- School of Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel
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2
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Dan X, Xu M, Yan Y, Shi Q. Generalized master equation for charge transport in a molecular junction: Exact memory kernels and their high order expansion. J Chem Phys 2022; 156:134114. [PMID: 35395901 DOI: 10.1063/5.0086663] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We derive a set of generalized master equations (GMEs) to study charge transport dynamics in molecular junctions using the Nakajima-Zwanzig-Mori projection operator approach. In the new GME, time derivatives of population on each quantum state of the molecule, as well as the tunneling current, are calculated as the convolution of time non-local memory kernels with populations on all system states. The non-Markovian memory kernels are obtained by combining the hierarchical equations of motion (HEOM) method and a previous derived Dyson relation for the exact kernel. A perturbative expansion of these memory kernels is then calculated using the extended HEOM developed in our previous work [M. Xu et al., J. Chem. Phys. 146, 064102 (2017)]. By using the resonant level model and the Anderson impurity model, we study properties of the exact memory kernels and analyze convergence properties of their perturbative expansions with respect to the system-bath coupling strength and the electron-electron repulsive energy. It is found that exact memory kernels calculated from HEOM exhibit short memory times and decay faster than the population and current dynamics. The high order perturbation expansion of the memory kernels can give converged results in certain parameter regimes. The Padé and Landau-Zener resummation schemes are also found to give improved results over low order perturbation theory.
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Affiliation(s)
- Xiaohan Dan
- Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences, Zhongguancun, Beijing 100190, China and University of Chinese Academy of Sciences, Beijing 100049, China
| | - Meng Xu
- Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences, Zhongguancun, Beijing 100190, China and University of Chinese Academy of Sciences, Beijing 100049, China
| | - Yaming Yan
- Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences, Zhongguancun, Beijing 100190, China and University of Chinese Academy of Sciences, Beijing 100049, China
| | - Qiang Shi
- Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences, Zhongguancun, Beijing 100190, China and University of Chinese Academy of Sciences, Beijing 100049, China
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3
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Ullah A, Han L, Yan YA, Zheng X, Yan Y, Chernyak V. Stochastic equation of motion approach to fermionic dissipative dynamics. II. Numerical implementation. J Chem Phys 2020; 152:204106. [DOI: 10.1063/1.5142166] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023] Open
Affiliation(s)
- Arif Ullah
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Lu Han
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Yun-An Yan
- School of Physics and Optoelectronic Engineering, Ludong University, Shandong 264025, China
| | - Xiao Zheng
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - YiJing Yan
- Hefei National Laboratory for Physical Sciences at the Microscale and iChEM, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Vladimir Chernyak
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
- Department of Chemistry, Wayne State University, 5101 Cass Avenue, Detroit, Michigan 48202, USA
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Cohen G, Galperin M. Green’s function methods for single molecule junctions. J Chem Phys 2020; 152:090901. [DOI: 10.1063/1.5145210] [Citation(s) in RCA: 25] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023] Open
Affiliation(s)
- Guy Cohen
- The Raymond and Beverley Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel
- School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel
| | - Michael Galperin
- Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, USA
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Zhang HD, Cui L, Gong H, Xu RX, Zheng X, Yan Y. Hierarchical equations of motion method based on Fano spectrum decomposition for low temperature environments. J Chem Phys 2020; 152:064107. [PMID: 32061227 DOI: 10.1063/1.5136093] [Citation(s) in RCA: 27] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The hierarchical equations of motion (HEOM) method has become one of the most popular methods for the studies of the open quantum system. However, its applicability to systems at ultra-low temperatures is largely restrained by the enormous computational cost, which is caused by the numerous exponential functions required to accurately characterize the non-Markovian memory of the reservoir environment. To overcome this problem, a Fano spectrum decomposition (FSD) scheme has been proposed recently [Cui et al., J. Chem. Phys. 151, 024110 (2019)], which expands the reservoir correlation functions using polynomial-exponential functions and hence greatly reduces the size of the memory basis set. In this work, we explicitly establish the FSD-based HEOM formalisms for both bosonic and fermionic environments. The accuracy and efficiency of the FSD-based HEOM are exemplified by the calculated low-temperature dissipative dynamics of a spin-boson model and the dynamic and static properties of a single-orbital Anderson impurity model in the Kondo regime. The encouraging numerical results highlight the practicality and usefulness of the FSD-based HEOM method for general open systems at ultra-low temperatures.
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Affiliation(s)
- Hou-Dao Zhang
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Lei Cui
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Hong Gong
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Rui-Xue Xu
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xiao Zheng
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - YiJing Yan
- Hefei National Laboratory for Physical Sciences at the Microscale, iChEM, University of Science and Technology of China, Hefei, Anhui 230026, China
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Ridley M, Gull E, Cohen G. Lead geometry and transport statistics in molecular junctions. J Chem Phys 2019; 150:244107. [DOI: 10.1063/1.5096244] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Michael Ridley
- School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel
- The Raymond and Beverley Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Emanuel Gull
- Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
- Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA
| | - Guy Cohen
- School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel
- The Raymond and Beverley Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel
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Schwarz F, Weymann I, von Delft J, Weichselbaum A. Nonequilibrium Steady-State Transport in Quantum Impurity Models: A Thermofield and Quantum Quench Approach Using Matrix Product States. PHYSICAL REVIEW LETTERS 2018; 121:137702. [PMID: 30312054 DOI: 10.1103/physrevlett.121.137702] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/28/2017] [Revised: 12/21/2017] [Indexed: 06/08/2023]
Abstract
The numerical renormalization group (NRG) is tailored to describe interacting impurity models in equilibrium, but it faces limitations for steady-state nonequilibrium, arising, e.g., due to an applied bias voltage. We show that these limitations can be overcome by describing the thermal leads using a thermofield approach, integrating out high energy modes using NRG, and then treating the nonequilibrium dynamics at low energies using a quench protocol, implemented using the time-dependent density matrix renormalization group. This yields quantitatively reliable results for the current (with errors ≲3%) down to the exponentially small energy scales characteristic of impurity models. We present results of benchmark quality for the temperature and magnetic field dependence of the zero-bias conductance peak for the single-impurity Anderson model.
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Affiliation(s)
- F Schwarz
- Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience, Ludwig-Maximilians-Universität, Theresienstraße 37, 80333 München, Germany
| | - I Weymann
- Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland
| | - J von Delft
- Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience, Ludwig-Maximilians-Universität, Theresienstraße 37, 80333 München, Germany
| | - A Weichselbaum
- Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience, Ludwig-Maximilians-Universität, Theresienstraße 37, 80333 München, Germany
- Department of Condensed Matter Physics and Materials Science, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
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Cheng Y, Li Z, Wei J, Nie Y, Yan Y. Transient dynamics of a quantum-dot: From Kondo regime to mixed valence and to empty orbital regimes. J Chem Phys 2018; 148:134111. [DOI: 10.1063/1.5013038] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/28/2022] Open
Affiliation(s)
- YongXi Cheng
- Department of Science, Taiyuan Institute of Technology, Taiyuan 030008, China
| | - ZhenHua Li
- State Key Laboratory of Structural Chemistry, Fujian Institute of Research on the Structure of Matter, Chinese Academy of Sciences, Fuzhou 350002, China
- Beijing Computational Science Research Center, Beijing 100084, China
| | - JianHua Wei
- Department of Physics, Renmin University of China, Beijing 100872, China
| | - YiHang Nie
- Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
| | - YiJing Yan
- Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
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Nghiem HTM, Costi TA. Time Evolution of the Kondo Resonance in Response to a Quench. PHYSICAL REVIEW LETTERS 2017; 119:156601. [PMID: 29077442 DOI: 10.1103/physrevlett.119.156601] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2017] [Indexed: 05/26/2023]
Abstract
We investigate the time evolution of the Kondo resonance in response to a quench by applying the time-dependent numerical renormalization group (TDNRG) approach to the Anderson impurity model in the strong correlation limit. For this purpose, we derive within the TDNRG approach a numerically tractable expression for the retarded two-time nonequilibrium Green function G(t+t^{'},t), and its associated time-dependent spectral function, A(ω,t), for times t both before and after the quench. Quenches from both mixed valence and Kondo correlated initial states to Kondo correlated final states are considered. For both cases, we find that the Kondo resonance in the zero temperature spectral function, a preformed version of which is evident at very short times t→0^{+}, only fully develops at very long times t≳1/T_{K}, where T_{K} is the Kondo temperature of the final state. In contrast, the final state satellite peaks develop on a fast time scale 1/Γ during the time interval -1/Γ≲t≲+1/Γ, where Γ is the hybridization strength. Initial and final state spectral functions are recovered in the limits t→-∞ and t→+∞, respectively. Our formulation of two-time nonequilibrium Green functions within the TDNRG approach provides a first step towards using this method as an impurity solver within nonequilibrium dynamical mean field theory.
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Affiliation(s)
- H T M Nghiem
- Peter Grünberg Institut and Institute for Advanced Simulation, Research Centre Jülich, 52425 Jülich, Germany
- Advanced Institute for Science and Technology, Hanoi University of Science and Technology, 10000 Hanoi, Vietnam
| | - T A Costi
- Peter Grünberg Institut and Institute for Advanced Simulation, Research Centre Jülich, 52425 Jülich, Germany
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Chen HT, Cohen G, Reichman DR. Inchworm Monte Carlo for exact non-adiabatic dynamics. I. Theory and algorithms. J Chem Phys 2017; 146:054105. [DOI: 10.1063/1.4974328] [Citation(s) in RCA: 34] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Hsing-Ta Chen
- Department of Chemistry, Columbia University, New York, New York 10027, USA
- The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel
| | - Guy Cohen
- The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel
- School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
| | - David R. Reichman
- Department of Chemistry, Columbia University, New York, New York 10027, USA
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Sayyad S, Eckstein M. Slowdown of the Electronic Relaxation Close to the Mott Transition. PHYSICAL REVIEW LETTERS 2016; 117:096403. [PMID: 27610867 DOI: 10.1103/physrevlett.117.096403] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2016] [Indexed: 06/06/2023]
Abstract
We investigate the time-dependent reformation of the quasiparticle peak in a correlated metal near the Mott transition, after the system is quenched into a hot electron state and equilibrates with an environment which is colder than the Fermi-liquid crossover temperature. Close to the transition, we identify a purely electronic bottleneck time scale, which depends on the spectral weight around the Fermi energy in the bad metallic phase in a nonlinear way. This time scale can be orders of magnitude larger than the bare and renormalized electronic hopping time, so that a separation of electronic and lattice time scales may break down. The results are obtained using nonequilibrium dynamical mean-field theory and a slave-rotor representation of the Anderson impurity model.
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Affiliation(s)
- Sharareh Sayyad
- Max Planck Institute for the Structure and Dynamics of Matter, 22761 Hamburg, Germany and University of Hamburg-CFEL, 22761 Hamburg, Germany
| | - Martin Eckstein
- Max Planck Institute for the Structure and Dynamics of Matter, 22761 Hamburg, Germany and University of Hamburg-CFEL, 22761 Hamburg, Germany
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