Abstract

Nonhyperbolicity, as characterized by the coexistence of Kolmogorov-Arnold-Moser (KAM) tori and chaos in the phase space, is generic in classical Hamiltonian systems. An open but fundamental question in physics concerns the relativistic quantum manifestations of nonhyperbolic dynamics. We choose the mushroom billiard that has been mathematically proven to be nonhyperbolic, and study the resonant tunneling dynamics of a massless Dirac fermion. We find that the tunneling rate as a function of the energy exhibits a striking "clustering" phenomenon, where the majority of the values of the rate concentrate on a narrow region, as a result of the chaos component in the classical phase space. Relatively few values of the tunneling rate, however, spread outside the clustering region due to the integrable component. Resonant tunneling of electrons in nonhyperbolic chaotic graphene systems exhibits a similar behavior. To understand these numerical results, we develop a theoretical framework by combining analytic solutions of the Dirac equation in certain integrable domains and physical intuitions gained from current understanding of the quantum manifestations of chaos. In particular, we employ a theoretical formalism based on the concept of self-energies to calculate the tunneling rate and analytically solve the Dirac equation in one dimension as well as in two dimensions for a circular-ring-type of tunneling systems exhibiting integrable dynamics in the classical limit. Because relatively few and distinct classical periodic orbits are present in the integrable component, the corresponding relativistic quantum states can have drastically different behaviors, leading to a wide spread in the values of the tunneling rate in the energy-rate plane. In contrast, the chaotic component has embedded within itself an infinite number of unstable periodic orbits, which provide far more quantum states for tunneling. Due to the nature of chaos, these states are characteristically similar, leading to clustering of the values of the tunneling rate in a narrow band. The appealing characteristic of our work is a demonstration and physical understanding of the "mixed" role played by chaos and regular dynamics in shaping relativistic quantum tunneling dynamics.

Original languageEnglish (US)
Article number224304
JournalPhysical Review B - Condensed Matter and Materials Physics
Volume87
Issue number22
DOIs
StatePublished - Jun 19 2013

Fingerprint

Fermions
Chaotic systems
Chaos theory
fermions
chaos
Resonant tunneling
Orbits
resonant tunneling
Dirac equation
Hamiltonians
Graphite
Graphene
orbits
Demonstrations
Physics
Electrons
narrowband
energy
graphene
formalism

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Electronic, Optical and Magnetic Materials

Cite this

Relativistic quantum tunneling of a Dirac fermion in nonhyperbolic chaotic systems. / Ni, Xuan; Huang, Liang; Ying, Lei; Lai, Ying-Cheng.

In: Physical Review B - Condensed Matter and Materials Physics, Vol. 87, No. 22, 224304, 19.06.2013.

Research output: Contribution to journalArticle

@article{c2c736ff66d2414585656e7cb580d4fa,
title = "Relativistic quantum tunneling of a Dirac fermion in nonhyperbolic chaotic systems",
abstract = "Nonhyperbolicity, as characterized by the coexistence of Kolmogorov-Arnold-Moser (KAM) tori and chaos in the phase space, is generic in classical Hamiltonian systems. An open but fundamental question in physics concerns the relativistic quantum manifestations of nonhyperbolic dynamics. We choose the mushroom billiard that has been mathematically proven to be nonhyperbolic, and study the resonant tunneling dynamics of a massless Dirac fermion. We find that the tunneling rate as a function of the energy exhibits a striking {"}clustering{"} phenomenon, where the majority of the values of the rate concentrate on a narrow region, as a result of the chaos component in the classical phase space. Relatively few values of the tunneling rate, however, spread outside the clustering region due to the integrable component. Resonant tunneling of electrons in nonhyperbolic chaotic graphene systems exhibits a similar behavior. To understand these numerical results, we develop a theoretical framework by combining analytic solutions of the Dirac equation in certain integrable domains and physical intuitions gained from current understanding of the quantum manifestations of chaos. In particular, we employ a theoretical formalism based on the concept of self-energies to calculate the tunneling rate and analytically solve the Dirac equation in one dimension as well as in two dimensions for a circular-ring-type of tunneling systems exhibiting integrable dynamics in the classical limit. Because relatively few and distinct classical periodic orbits are present in the integrable component, the corresponding relativistic quantum states can have drastically different behaviors, leading to a wide spread in the values of the tunneling rate in the energy-rate plane. In contrast, the chaotic component has embedded within itself an infinite number of unstable periodic orbits, which provide far more quantum states for tunneling. Due to the nature of chaos, these states are characteristically similar, leading to clustering of the values of the tunneling rate in a narrow band. The appealing characteristic of our work is a demonstration and physical understanding of the {"}mixed{"} role played by chaos and regular dynamics in shaping relativistic quantum tunneling dynamics.",
author = "Xuan Ni and Liang Huang and Lei Ying and Ying-Cheng Lai",
year = "2013",
month = "6",
day = "19",
doi = "10.1103/PhysRevB.87.224304",
language = "English (US)",
volume = "87",
journal = "Physical Review B-Condensed Matter",
issn = "0163-1829",
publisher = "American Institute of Physics Publising LLC",
number = "22",

}

TY - JOUR

T1 - Relativistic quantum tunneling of a Dirac fermion in nonhyperbolic chaotic systems

AU - Ni, Xuan

AU - Huang, Liang

AU - Ying, Lei

AU - Lai, Ying-Cheng

PY - 2013/6/19

Y1 - 2013/6/19

N2 - Nonhyperbolicity, as characterized by the coexistence of Kolmogorov-Arnold-Moser (KAM) tori and chaos in the phase space, is generic in classical Hamiltonian systems. An open but fundamental question in physics concerns the relativistic quantum manifestations of nonhyperbolic dynamics. We choose the mushroom billiard that has been mathematically proven to be nonhyperbolic, and study the resonant tunneling dynamics of a massless Dirac fermion. We find that the tunneling rate as a function of the energy exhibits a striking "clustering" phenomenon, where the majority of the values of the rate concentrate on a narrow region, as a result of the chaos component in the classical phase space. Relatively few values of the tunneling rate, however, spread outside the clustering region due to the integrable component. Resonant tunneling of electrons in nonhyperbolic chaotic graphene systems exhibits a similar behavior. To understand these numerical results, we develop a theoretical framework by combining analytic solutions of the Dirac equation in certain integrable domains and physical intuitions gained from current understanding of the quantum manifestations of chaos. In particular, we employ a theoretical formalism based on the concept of self-energies to calculate the tunneling rate and analytically solve the Dirac equation in one dimension as well as in two dimensions for a circular-ring-type of tunneling systems exhibiting integrable dynamics in the classical limit. Because relatively few and distinct classical periodic orbits are present in the integrable component, the corresponding relativistic quantum states can have drastically different behaviors, leading to a wide spread in the values of the tunneling rate in the energy-rate plane. In contrast, the chaotic component has embedded within itself an infinite number of unstable periodic orbits, which provide far more quantum states for tunneling. Due to the nature of chaos, these states are characteristically similar, leading to clustering of the values of the tunneling rate in a narrow band. The appealing characteristic of our work is a demonstration and physical understanding of the "mixed" role played by chaos and regular dynamics in shaping relativistic quantum tunneling dynamics.

AB - Nonhyperbolicity, as characterized by the coexistence of Kolmogorov-Arnold-Moser (KAM) tori and chaos in the phase space, is generic in classical Hamiltonian systems. An open but fundamental question in physics concerns the relativistic quantum manifestations of nonhyperbolic dynamics. We choose the mushroom billiard that has been mathematically proven to be nonhyperbolic, and study the resonant tunneling dynamics of a massless Dirac fermion. We find that the tunneling rate as a function of the energy exhibits a striking "clustering" phenomenon, where the majority of the values of the rate concentrate on a narrow region, as a result of the chaos component in the classical phase space. Relatively few values of the tunneling rate, however, spread outside the clustering region due to the integrable component. Resonant tunneling of electrons in nonhyperbolic chaotic graphene systems exhibits a similar behavior. To understand these numerical results, we develop a theoretical framework by combining analytic solutions of the Dirac equation in certain integrable domains and physical intuitions gained from current understanding of the quantum manifestations of chaos. In particular, we employ a theoretical formalism based on the concept of self-energies to calculate the tunneling rate and analytically solve the Dirac equation in one dimension as well as in two dimensions for a circular-ring-type of tunneling systems exhibiting integrable dynamics in the classical limit. Because relatively few and distinct classical periodic orbits are present in the integrable component, the corresponding relativistic quantum states can have drastically different behaviors, leading to a wide spread in the values of the tunneling rate in the energy-rate plane. In contrast, the chaotic component has embedded within itself an infinite number of unstable periodic orbits, which provide far more quantum states for tunneling. Due to the nature of chaos, these states are characteristically similar, leading to clustering of the values of the tunneling rate in a narrow band. The appealing characteristic of our work is a demonstration and physical understanding of the "mixed" role played by chaos and regular dynamics in shaping relativistic quantum tunneling dynamics.

UR - http://www.scopus.com/inward/record.url?scp=84879731499&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84879731499&partnerID=8YFLogxK

U2 - 10.1103/PhysRevB.87.224304

DO - 10.1103/PhysRevB.87.224304

M3 - Article

VL - 87

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

SN - 0163-1829

IS - 22

M1 - 224304

ER -