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| In physics, '''Berry connection''' and '''Berry curvature''' are related concepts, which can be viewed, respectively, as a local gauge potential and gauge field associated with the [[Berry phase]]. These concepts were introduced by [[Michael Berry (physicist)|Michael Berry]] in a paper published in 1984<ref>
| | Most old school beat 'em ups tend to lack anything that even resembles depth. Little [http://ad4Game.com/ Fighter] 2 characteristics combo attacks, counterattacks, transformations, and other nifty things that include a great deal of replay value. Again, Little Fighter 2' play mechanisms remind me lots of River City Ransom. That sport was ahead of it is time, that's for sure.<br><br>Follow up existing customers. That is a great time to re-connect with your great customers and continue to build your relationship together and their [http://Xda-developers.com/ loyalty] for you. Remind them you exist to assist and get feedback on which they want/need. Perhaps a specific offer or discount during this time will build company and loyalty to your own business name.<br><br>Kim apologized. A lot. Then she apologized some more. In a attempt to make up for our irritation, she said she would upgrade us both to suites, and again, she was sorry. She scribbled two-room figures on two new important envelopes and handed them to us. Her penmanship was deteriorating badly.<br><br>Another step in healthy eating habits is consuming organic vegetables and fruits. These meals have more vitamins and minerals coming from earth that is not depleted by erosion and mono-cropping. Artificial pesticides and herbicides are not used in the creation of natural fruits and, and they don't allow the utilization of the artificial pesticides and herbicides which are used in traditional growing systems. Believe it - limiting the intake of pesticides and fertilizers is good for your own insides!<br><br>Use the KISS (Keep It Simple Silly) method. Regular walking, weight lifting, eating a healthful diet (see number 4) and drinking tons of water are the key elements to living a healthy life style.<br><br>Fold the armholes and pin. Fe to hold in place and attach the binding or ribbon beginning in the bottom of the armhole. Allow extra binding or ribbon to extend past the armholes and use to tie a bow at the shoulders.<br><br>Part of being prepared is having all of your gear is in working order before you leave the last rehearsal. There will be times when ill luck throws you a curve, but 99% of onstage gear malfunctions are avoidable. Give everything the once-over ahead of time and prepare yourself to field any emergencies with additional strings, spare cables, fuses, AC adapters and duct tape. Great lord, don't forget the duck tape.<br><br>Help your child brush and floss their teeth up to the age of six years aged. They will need your help in understanding how to properly brush and floss their teeth. Also, be a great example to your child by regularly brushing and flossing your teeth If you have any sort of questions concerning where and how to utilize [http://opisy-gg.info.pl http://opisy-gg.info.pl], you could call us at our website. . |
| {{cite journal
| |
| |author = Berry, M. V.
| |
| |title = Quantal Phase Factors Accompanying Adiabatic Changes
| |
| |volume = 392
| |
| |number = 1802
| |
| |pages = 45–57
| |
| |year = 1984
| |
| |doi = 10.1098/rspa.1984.0023
| |
| |url = http://rspa.royalsocietypublishing.org/content/392/1802/45.abstract
| |
| |eprint = http://rspa.royalsocietypublishing.org/content/392/1802/45.full.pdf+html
| |
| |journal = [[Proceedings of the Royal Society A]]
| |
| |bibcode = 1984RSPSA.392...45B }}</ref>
| |
| emphasizing how [[geometric phase]]s provide a powerful unifying concept in several branches of [[Classical physics|classical]] and [[quantum physics]]. Such phases have come to be known as [[geometric phase|Berry phases]].
| |
| | |
| ==Berry phase and cyclic adiabatic evolution==
| |
| | |
| In quantum mechanics, the Berry phase arises in a cyclic [[Adiabatic theorem|adiabatic]] evolution. The quantum [[adiabatic theorem]] applies to a system whose [[Hamiltonian (quantum mechanics)|Hamiltonian]] <math>H(\mathbf R)</math> depends on a (vector) parameter <math>\mathbf R</math> that varies with time <math>t</math>. If the <math>n</math>'th [[eigenvalue]] <math>\varepsilon_n(\mathbf R)</math> remains non-degenerate everywhere along the path and the variation with time ''t'' is sufficiently slow, then a system initially in the [[eigenstate]]
| |
| <math>\, |n(\mathbf R(0))\rangle </math> will remain in an instantaneous eigenstate <math>\, |n(\mathbf R(t))\rangle </math> of the Hamiltonian <math>\, H(\mathbf R(t))</math>, up to a phase, throughout the process. Regarding the phase, the state at time ''t'' can be written as<ref name=Sakurai>
| |
| {{cite book
| |
| |author= Sakurai, J.J.
| |
| |title= Modern Quantum Mechanics
| |
| |year=2005
| |
| |publisher=Addison–Wesley
| |
| |volume= Revised Edition
| |
| |url= http://books.google.com/books?id=w2a8QgAACAAJ}}
| |
| </ref>
| |
| :<math>
| |
| |\Psi_n(t)\rangle =e^{i\gamma_n(t)}\,
| |
| e^{-{i\over\hbar}\int_0 ^t dt'\varepsilon_n(\mathbf R(t'))}\,
| |
| | n(\mathbf R(t))\rangle,
| |
| </math>
| |
| where the second exponential term is the "dynamic phase factor." The first exponential term is the geometric term, with <math>\gamma_n</math> being the Berry phase. By plugging into the [[Schr%C3%B6dinger_equation#Time_dependent_equation|time-dependent Schrödinger equation]], it can be shown that
| |
| :<math>
| |
| \gamma_n(t)=i\int_0^t dt'\,\langle n(\mathbf R(t'))|{d\over dt'}|n(\mathbf R(t'))\rangle=i\int_{\mathbf R(0)}^{\mathbf R(t)} d\mathbf R\,\langle n(\mathbf R)|\nabla_{\mathbf R}|n(\mathbf R)\rangle,
| |
| </math>
| |
| indicating that the Berry phase only depends on the path in the parameter space, not on the rate at which the path is traversed.
| |
| | |
| In the case of a cyclic evolution around a closed path <math>\mathcal C</math> such that <math>\mathbf R(T)=\mathbf R(0)</math>, the closed-path Berry phase is
| |
| :<math>
| |
| \gamma_n=i\oint_{\mathcal C} d\mathbf R\,\langle n(\mathbf R)|\nabla_{\mathbf R}|n(\mathbf R)\rangle.
| |
| </math>
| |
| An example of physical system where an electron moves along a closed path is cyclotron motion (details are given in the page of [[Geometric_phase|Berry phase]]). Berry phase must be considered to obtain the correct quantization condition.
| |
| | |
| | |
| ==Gauge transformation==
| |
| | |
| Without changing the physics, we can make a [[Gauge theory#Mathematical formalism|gauge transformation]]
| |
| :<math>
| |
| |\tilde n(\mathbf R)\rangle=e^{-i\beta(\mathbf R)}|n(\mathbf R)\rangle
| |
| </math>
| |
| to a new set of states that differ from the original ones only by an <math>\mathbf R</math>-dependent phase factor. This modifies the open-path Berry phase to be <math>\tilde\gamma_n(t)=\gamma_n(t)+\beta(t)-\beta(0)</math>. For a closed path, continuity requires that <math>\beta(T)-\beta(0)=2\pi m</math> (<math>m</math> an integer), and it follows that <math>\gamma_n</math> is invariant, modulo <math>2\pi</math>, under an arbitrary gauge transformation. | |
| | |
| ==Berry connection==
| |
| The closed-path Berry phase defined above can be expressed as
| |
| :<math> | |
| \gamma_n=\int_\mathcal{C} d\mathbf R\cdot \mathcal{A}_n(\mathbf R)
| |
| </math>
| |
| where
| |
| :<math>
| |
| \mathcal{A}_n(\mathbf R)=i\langle n(\mathbf R)|\nabla_{\mathbf R}|n(\mathbf R)\rangle.
| |
| </math>
| |
| is a vector-valued function known as the Berry connection (or Berry potential). The Berry connection is gauge-dependent, transforming as
| |
| <math>\tilde{\mathcal{A}}_n(\mathbf R)=\mathcal{A}_n (\mathbf R)+\nabla_{\mathbf R\,}\beta(\mathbf R)</math>. Hence the local Berry connection <math>\mathcal{A}_n(\mathbf R)</math> can never be physically observable. However, its integral along a closed path, the Berry phase <math>\gamma_n</math>, is gauge-invariant up to an integer multiple of <math>2\pi</math>. Thus, <math>e^{i\gamma_n}</math> is absolutely gauge-invariant, and may be related to physical observables.
| |
| | |
| ==Berry curvature==
| |
| | |
| The Berry curvature is an [[anti-symmetric]] second-rank tensor derived from the Berry connection via
| |
| :<math>
| |
| \Omega_{n,\mu\nu} (\mathbf R)={\partial\over\partial R^\mu}\mathcal{A}_{n,\nu}(\mathbf R)-{\partial\over\partial R^\nu}\mathcal{A}_{n,\mu}(\mathbf R).
| |
| </math>
| |
| In a three-dimensional parameter space the Berry curvature can be written in the [[pseudovector]] form | |
| :<math>
| |
| \mathbf\Omega_n(\mathbf R)=\nabla_{\mathbf R} \times\mathcal{A}_n(\mathbf R).
| |
| </math>
| |
| The tensor and pseduovector forms of the Berry curvature are related to each other through the [[Levi-Civita symbol|Levi-Civita]] antisymmetric tensor as
| |
| <math>\Omega_{n,\mu\nu}=\epsilon_{\mu\nu\xi}\,\mathbf\Omega_{n,\xi}</math>. In contrast to the Berry connection, which is physical only after integrating around a closed path, the Berry curvature is a gauge-invariant local manifestation of the geometric properties of the wavefunctions in the parameter space, and has proven to be an essential physical ingredient for understanding a variety of electronic properties.<ref name=resta2000>
| |
| {{cite journal
| |
| |title = Manifestations of Berry's phase in molecules and in condensed matter
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| |author = Resta, Raffaele
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| |journal = J. Phys.: Condens. Matter
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| |volume = 12
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| |pages = R107
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| |year = 2000
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| |doi = 10.1088/0953-8984/12/9/201
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| |bibcode = 2000JPCM...12R.107R }}
| |
| </ref><ref name=BerryMod>
| |
| {{cite journal
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| |title = Berry phase effects on electronic properties
| |
| |author = Xiao, Di
| |
| |author2 = Chang, Ming-Che
| |
| |author3 = Niu, Qian
| |
| |journal = Rev. Mod. Phys.
| |
| |volume = 82
| |
| |number = 3
| |
| |pages = 1959–2007
| |
| |numpages = 48
| |
| |date=Jul 2010
| |
| |doi = 10.1103/RevModPhys.82.1959
| |
| |publisher = American Physical Society
| |
| |bibcode=2010RvMP...82.1959X
| |
| |arxiv = 0907.2021 }}
| |
| </ref>
| |
| | |
| For a closed path <math>\mathcal C</math> that forms the boundary of a surface <math>\mathcal{S}</math>, the closed-path Berry phase can be rewritten using [[Stokes' theorem]] as
| |
| :<math>
| |
| \gamma_n=\int_\mathcal{S} d\mathbf S\cdot\mathbf\Omega_n (\mathbf R).
| |
| </math>
| |
| If the surface is a closed manifold, the boundary term vanishes, but the indeterminacy of the boundary term modulo <math>2\pi</math> manifests itself in the [[Chern–Gauss–Bonnet theorem|Chern theorem]], which states that the integral of the Berry curvature over a closed manifold is quantized in units of <math>2\pi</math>. This number is the so-called [[Chern number]], and is essential for understanding various quantization effects.
| |
| | |
| Finally, note that the Berry curvature can also be written, using [[perturbation theory]], as a sum over all other eigenstates in the form
| |
| :<math>
| |
| \Omega_{n,\mu\nu}(\mathbf R)=i\sum_{n'\neq n}{\langle n|(\partial H/\partial R_\mu) |n'\rangle\langle n'|(\partial H/\partial R_\nu) | n\rangle-(\nu\leftrightarrow\mu)\over(\varepsilon_n-\varepsilon_{n'})^2}.
| |
| </math>
| |
| | |
| ==Example: Spinor in a magnetic field==
| |
| | |
| The Hamiltonian of a spin-1/2 particle in a [[magnetic field]] can be written as<ref name=Sakurai/>
| |
| :<math>
| |
| H=\mu\mathbf\sigma\cdot\mathbf B,
| |
| </math>
| |
| where <math>\mathbf\sigma</math> denote the [[Pauli matrices]], <math>\mu</math> is the [[magnetic moment]], and '''B''' is the magnetic field. In three dimensions, the eigenstates have energies <math>\pm\mu B</math> and their eigenvectors are
| |
| :<math>
| |
| |u_-\rangle=
| |
| \begin{pmatrix}
| |
| \sin{\theta\over 2}e^{-i\phi}\\
| |
| -\cos{\theta\over 2}
| |
| \end{pmatrix},
| |
| |u_+\rangle=
| |
| \begin{pmatrix}
| |
| \cos{\theta\over 2}e^{-i\phi}\\
| |
| \sin{\theta\over 2}
| |
| \end{pmatrix}.
| |
| </math>
| |
| Now consider the <math>|u_-\rangle</math> state. Its Berry connection can be computed as
| |
| <math>\mathcal{A}_\theta=\langle u|i\partial_\theta u\rangle=0,
| |
| </math>
| |
| <math>
| |
| \mathcal{A}_\phi=\langle u|i\partial_\phi u\rangle=\sin^2{\theta\over 2}
| |
| </math>,
| |
| and the Berry curvature is | |
| <math>
| |
| \Omega_{\theta\phi}=\partial_\theta\mathcal{A}_\phi-\partial_\phi\mathcal A_\theta={1\over 2}\sin\theta.
| |
| </math>
| |
| If we choose a new gauge by multiplying <math>|u_-\rangle</math> by <math>e^{i\phi}</math>, the Berry connections are
| |
| <math>\mathcal{A}_\theta=0</math> and <math>\mathcal{A}_\phi=-\cos^2{\theta\over 2}</math>, while the Berry curvature remains the same. This is consistent with the conclusion that the Berry connection is gauge-dependent while the Berry curvature is not.
| |
| | |
| The Berry curvature per solid angle is given by <math>\overline{\Omega}_{\theta\phi}=\Omega_{\theta\phi}/\sin\theta=1/2</math>. In this case, the Berry phase corresponding to any given path on the unit sphere <math>\mathcal S^2</math> in magnetic-field space is just half the solid angle subtended by the path.
| |
| The integral of the Berry curvature over the whole sphere is therefore exactly <math>2\pi</math>, so that the Chern number is unity, consistent with the Chern theorem.
| |
| | |
| ==Applications in crystals==
| |
| | |
| The Berry phase plays an important role in modern investigations of electronic properties in crystalline solids<ref name=BerryMod/> and in the theory of the [[quantum_Hall_effect#Mathematics|quantum Hall effect]].<ref>
| |
| {{cite journal
| |
| |title = Quantized Hall Conductance in a Two-Dimensional Periodic Potential
| |
| |author = Thouless, D. J.
| |
| |author2 = Kohmoto, M.
| |
| |author2 = Nightingale, M. P.
| |
| |author2 = den Nijs, M.
| |
| |journal = Phys. Rev. Lett.
| |
| |volume = 49
| |
| |number = 6
| |
| |pages = 405–408
| |
| |numpages = 3
| |
| |date=Aug 1982
| |
| |doi = 10.1103/PhysRevLett.49.405
| |
| |publisher = American Physical Society
| |
| |bibcode=1982PhRvL..49..405T
| |
| }}
| |
| </ref> | |
| The periodicity of the crystalline potential allows the application of the [[Bloch theorem]], which states that the Hamiltonian eigenstates take the form
| |
| :<math>
| |
| \psi_{n\mathbf k}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{n\mathbf k}(\mathbf r),
| |
| </math>
| |
| where <math>n</math> is a band index, <math>\mathbf k</math> is a wavevector in the [[reciprocal space|reciprocal-space]] ([[Brillouin zone]]), and <math>u_{n\mathbf k}(\mathbf r)</math> is a periodic function of <math>\mathbf r</math>. Then, letting <math>\mathbf k</math> play the role of the parameter <math>\mathbf R</math>, one can define Berry phases, connections, and curvatures in the reciprocal space. For example, the Berry connection in reciprocal space is
| |
| :<math>
| |
| \mathcal{A}_n(\mathbf k)=i\langle n(\mathbf k)|\nabla_{\mathbf k}|n(\mathbf k)\rangle.
| |
| </math> | |
| Because the Bloch theorem also implies that the reciprocal space itself is closed, with the Brillouin zone having the topology of a 3-torus in three dimensions, the requirements of integrating over a closed loop or manifold can easily be satisfied. In this way, such properties as the [[electric polarization]], orbital [[magnetization]], [[Hall_effect#Anomalous_Hall_effect|anomalous Hall conductivity]], and orbital magnetoelectric coupling can be expressed in terms of Berry phases, connections, and curvatures.<ref name=BerryMod/><ref>
| |
| {{cite journal
| |
| |author=Chang, Ming-Che
| |
| |author2=Niu, Qian
| |
| |title=Berry curvature, orbital moment, and effective quantum theory of electrons in electromagnetic fields
| |
| |journal=Journal of Physics: Condensed Matter
| |
| |volume=20
| |
| |number=19
| |
| |pages=193202
| |
| |url=http://stacks.iop.org/0953-8984/20/i=19/a=193202
| |
| |year=2008}}
| |
| </ref><ref name=resta2010>
| |
| {{cite journal
| |
| |title = Electrical polarization and orbital magnetization: the modern theories
| |
| |author = Resta, Raffaele
| |
| |journal = J. Phys.: Condens. Matter
| |
| |volume = 22
| |
| |pages = 123201
| |
| |year = 2010
| |
| |doi = 10.1088/0953-8984/22/12/123201
| |
| |bibcode = 2010JPCM...22l3201R }}
| |
| </ref>
| |
| | |
| ==References==
| |
| | |
| {{reflist}}
| |
| | |
| ==External links==
| |
| *"[http://books.google.com/books?hl=en&lr=&id=5jOvlny96AkC&oi=fnd&pg=PR5&dq=+geometric+phases+shapere&ots=IoFIBvrl3O&sig=TKqsDeqR44j3q8fq0vieNlx1Foo#v=onepage&q&f=false The quantum phase, five years after.]" by M. Berry.
| |
| *"[http://www.physics.rutgers.edu/~dhv/talks/rahman.pdf Berry Phases and Curvatures in Electronic Structure Theory.]" A talk by D. Vanderbilt.
| |
| *"[http://www.physics.rutgers.edu/~dhv/talks/psik-vanderbilt-2010.pdf Berry-ology, Orbital Magnetolectric Effects, and Topological Insulators]" A talk by D. Vanderbilt.
| |
| | |
| [[Category:Quantum mechanics]]
| |
| [[Category:Classical mechanics]]
| |
| [[Category:Quantum phases]]
| |
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Another step in healthy eating habits is consuming organic vegetables and fruits. These meals have more vitamins and minerals coming from earth that is not depleted by erosion and mono-cropping. Artificial pesticides and herbicides are not used in the creation of natural fruits and, and they don't allow the utilization of the artificial pesticides and herbicides which are used in traditional growing systems. Believe it - limiting the intake of pesticides and fertilizers is good for your own insides!
Use the KISS (Keep It Simple Silly) method. Regular walking, weight lifting, eating a healthful diet (see number 4) and drinking tons of water are the key elements to living a healthy life style.
Fold the armholes and pin. Fe to hold in place and attach the binding or ribbon beginning in the bottom of the armhole. Allow extra binding or ribbon to extend past the armholes and use to tie a bow at the shoulders.
Part of being prepared is having all of your gear is in working order before you leave the last rehearsal. There will be times when ill luck throws you a curve, but 99% of onstage gear malfunctions are avoidable. Give everything the once-over ahead of time and prepare yourself to field any emergencies with additional strings, spare cables, fuses, AC adapters and duct tape. Great lord, don't forget the duck tape.
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