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In classical and quantum mechanics, the geometric phase, Pancharatnam–Berry phase (named after S. Pancharatnam and Sir Michael Berry), Pancharatnam phase or most commonly Berry phase, is a phase acquired over the course of a cycle, when the system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian. The phenomenon was first discovered in 1956, and rediscovered in 1984. It can be seen in the Aharonov–Bohm effect and in the conical intersection of potential energy surfaces. In the case of the Aharonov–Bohm effect, the adiabatic parameter is the magnetic field enclosed by two interference paths, and it is cyclic in the sense that these two paths form a loop. In the case of the conical intersection, the adiabatic parameters are the molecular coordinates. Apart from quantum mechanics, it arises in a variety of other wave systems, such as classical optics. As a rule of thumb, it occurs whenever there are at least two parameters affecting a wave, in the vicinity of some sort of singularity or hole in the topology.

Waves are characterized by amplitude and phase, and both may vary as a function of those parameters. The Berry phase occurs when both parameters are changed simultaneously but very slowly (adiabatically), and eventually brought back to the initial configuration. In quantum mechanics, this could involve rotations but also translations of particles, which are apparently undone at the end. Intuitively one expects that the waves in the system return to the initial state, as characterized by the amplitudes and phases (and accounting for the passage of time). However, if the parameter excursions correspond to a cyclic loop instead of a self-retracing back-and-forth variation, then it is possible that the initial and final states differ in their phases. This phase difference is the Berry phase, and its occurrence typically indicates that the system's parameter dependence is singular (undefined) for some combination of parameters.

To measure the Berry phase in a wave system, an interference experiment is required. The Foucault pendulum is an example from classical mechanics that is sometimes used to illustrate the Berry phase. This mechanics analogue of the Berry phase is known as the Hannay angle.

Theory

In general the geometric phase is given by:

$\gamma [C]=i{\oint }_C⟨n,t|\left({\mathrm{\nabla }}_R\right|n,t⟩)dR$

where $R$ parametrizes the cyclic adiabatic process. It follows a closed path $C$ in the appropriate parameter space. A recent review on the Berry phase effects on electronic properties was given by Xiao, Chang and Niu. Geometric phase along the closed path $C$ can also be calculated by integrating the Berry curvature over surface enclosed by $C$.

Examples of geometric phases

The Foucault pendulum

One of the easiest examples is the Foucault pendulum. An easy explanation in terms of geometric phases is given by von Bergmann and von Bergmann:

How does the pendulum precess when it is taken around a general path C? For transport along the equator, the pendulum will not precess. [...] Now if C is made up of geodesic segments, the precession will all come from the angles where the segments of the geodesics meet; the total precession is equal to the net deficit angle which in turn equals the solid angle enclosed by C modulo 2π. Finally, we can approximate any loop by a sequence of geodesic segments, so the most general result (on or off the surface of the sphere) is that the net precession is equal to the enclosed solid angle.

In summary, there are no inertial forces that could make the pendulum precess. Thus the orientation of the pendulum undergoes parallel transport along the path of fixed latitude. By the Gauss–Bonnet theorem the phase shift is given by the enclosed solid angle.

Polarized light in an optical fiber

A second example is linearly polarized light entering a single-mode optical fiber. Suppose the fiber traces out some path in space and the light exits the fiber in the same direction as it entered. Then compare the initial and final polarizations. In semiclassical approximation the fiber functions as a waveguide and the momentum of the light is at all times tangent to the fiber. The polarization can be thought of as an orientation perpendicular to the momentum. As the fiber traces out its path, the momentum vector of the light traces out a path on the sphere in momentum space. The path is closed since initial and final directions of the light coincide, and the polarization is a vector tangent to the sphere. Going to momentum space is equivalent to taking the Gauss map. There are no forces that could make the polarization turn, just the constraint to remain tangent to the sphere. Thus the polarization undergoes parallel transport and the phase shift is given by the enclosed solid angle (times the spin, which in case of light is 1).

Stochastic pump effect

A stochastic pump is a classical stochastic system that responds with nonzero, on average, currents to periodic changes of parameters. The stochastic pump effect can be interpreted in terms of a geometric phase in evolution of the moment generating function of stochastic currents.

Geometric phase defined on attractors

While Berry's phase was originally defined for linear Hamiltonian systems, it was soon realized by Ning and Haken that similar geometric phase can be defined for entirely different systems such as nonlinear dissipative systems that possess certain cyclic attractors. They showed that such cyclic attractors exist in a class of nonlinear dissipative systems with certain symmetries.

Exposing Berry/Pancharatnam phases in molecular adiabatic potential surface intersections

There are several ways to compute the Berry phase in molecules within the Born Oppenheimer framework. One way is through the "non-adiabatic coupling $M\times M$ matrix" defined by

${\tau }_{ij}^{\mu }=⟨{\psi }_i|{\partial }^{\mu }{\psi }_j⟩$

where ${\psi }_i$ is the adiabatic electronic wave function, depending on the nuclear parameters $R_{\mu }$. The nonadiabatic coupling can be used to define a loop integral, analogous to a Wilson loop (1974) in field theory, developed independently for molecular framework by M. Baer (1975, 1980, 2000). Given a closed loop $\mathrm{\Gamma }$, parameterized by $R_{\mu }\left(t\right)$ where $t\in [0,1]$ is a parameter and $R_{\mu }(t+1)=R_{\mu }\left(t\right)$. The D-matrix is given by:

$D\left[\mathrm{\Gamma }\right]=\hat{P}{e}^{{\oint }_{\mathrm{\Gamma }}{\tau }^{\mu }d{R}_{\mu }}$

(here, $\hat{P}$ is a path ordering symbol). It can be shown that once $M$ is large enough (i.e. a sufficient number of electronic states is considered) this matrix is diagonal with the diagonal elements equal to $e^{i{\beta }_j}$ where ${\beta }_j$ are the Berry phases associated with the loop for the $j$ adiabatic electronic state.

For time-reversal symmetrical electronic Hamiltonians the Berry phase reflects the number of conical intersections encircled by the loop. More accurately:

$e^{i{\beta }_j}={(-1)}^{N_j}$

where $N_j$ is the number of conical intersections involving the adiabatic state ${\psi }_j$ encircled by the loop $\mathrm{\Gamma }$.

An alternative to the D-matrix approach would be a direct calculation of the Pancharatnam phase. This is especially useful if one is interested only in the Berry phases of a single adiabatic state. In this approach, one takes a number $N+1$ of points $(n=0,...,N)$ along the loop $R\left(t_n\right)$ with $t_0=0$ and $t_N=1$ then using only the jth adiabatic states ${\psi }_j\left[R\left(t_n\right)\right]$ computes the Pancharatnam product of overlaps:

$I_j(\mathrm{\Gamma },N)=\prod _{n=0}^{N-1}⟨{\psi }_j\left[R\left(t_n\right)\right]|{\psi }_j\left[R\left(t_{n+1}\right)\right]⟩$

In the limit $N\to \mathrm{\infty }$ one has (See Ryb & Baer 2004 for explanation and some applications):

$I_j(\mathrm{\Gamma },N)\to e^{i{\beta }_j}$

Berry's phase and quantization of cyclotron motion

Electron subjected to magnetic field $B$ moves on a circular (cyclotron) orbit.[1] Classicaly, any cyclotron radius $R_c$ is acceptable. Quantum-mechanically, only discrete energy levels (Landau levels) are allowed and since $R_c$ is related to electron's energy, this corresponds to quantized values of $R_c$. The energy quantization condition obtained by solving Schrödinger's equation reads, for example, $E=(n+\alpha )\hslash {\omega }_c,\alpha =1/2$ for free electrons (in vacuum) or $E=v\sqrt{2(n+\alpha )eB\hslash },\alpha =0$ for electrons in graphene where $n=0,1,2,\dots$.[2] Although the derivation of these results is not difficult, there is an alternative way of deriving them which offers in some respect better physical insight into the Landau level quantization. This alternative way is based on the semiclassical Bohr-Sommerfeld quantization condition

$\hslash \oint d\mathbf{r}\cdot \mathbf{k}-e\oint d\mathbf{r}\cdot \mathbf{A}+\hslash \gamma =2\pi \hslash (n+1/2)$

which includes the Berry phase $\gamma$ picked up by the electron while it executes its (real-space) motion along the closed loop of the cyclotron orbit. These well-known facts have recently been nicely summarized by Jiamin Xue. For free electrons, $\gamma =0$ while $\gamma =\pi$ for electrons in graphene. It turns out that Berry's phase is directly linked to $\alpha =1/2$ of free electrons and $\alpha =0$ of electrons in graphene.

See also

• For the connection to mathematics, see curvature tensor,
• Adiabatic theorem,
• Aharonov–Bohm effect,
• Berry connection and curvature,
• Conical intersections of potential energy surfaces.
• Chern class
• Holonomy
• Hannay angle
• Wilson loop
• Winding number

Footnotes

• 1. For simplicity, we consider electrons confined to a plane, such as 2DEG and magnetic field perpendicular to the plane.

• 2. ${\omega }_c=eB/m$ is the cyclotron frequency (for free electrons) and $v$ is the Fermi velocity (of electrons in graphene).

Notes

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References

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  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

• V. Cantoni and L. Mistrangioli (1992) "Three-Point Phase, Symplectic Measure and Berry Phase", International Journal of Theoretical Physics vol. 31 p. 937.

• Richard Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications (Mathematical Surveys and Monographs, Volume 91), (2002) American Mathematical Society, ISBN 0-8218-1391-9. (See chapter 13 for a mathematical treatment)

• Connections to other physical phenomena (such as the Jahn–Teller effect) are discussed here: [1]

• Paper by Prof. Galvez at Colgate University, describing Geometric Phase in Optics: [2]

• Surya Ganguli, Fibre Bundles and Gauge Theories in Classical Physics: A Unified Description of Falling Cats, Magnetic Monopoles and Berry's Phase [3]

• Robert Batterman, Falling Cats, Parallel Parking, and Polarized Light [4]

• M. Baer, "Adiabatic and diabatic representations for atom-molecule collisions: treatment of the collinear arrangement", Chem. Phys. Lett. 35, 112 (1975); M. Baer, "Electronic non-adiabatic transitions: Derivation of the general adiabatic-diabatic transformation matrix", Mol. Phys. 40, 1011 (1980); M. Baer, "Existence of diabetic potentials and the quantization of the nonadiabatic matrix", J. Phys. Chem. A 104, 3181-3184 (2000).

• I. Ryb and R. Baer, "Combinatorial invariants and covariants as tools for conical intersections", J. Chem. Phys. 121, 10370-10375 (2004). R. Baer, "Born-Oppenheimer invariants along nuclear paths", J. Chem. Phys. 117, 7405 (2002).

• Frank Wilczek and Alfred Shapere, "Geometric Phases in Physics", World Scientific, 1989

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang