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Non-commutative mechanics, in mathematical & in condensed matter physics

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Symmetry,Integrability and Geometry:Methods and Applications Vol.2(2006),Paper 090,9pages Non-Commutative Mechanics in Mathematical &in Condensed Matter Physics ?Peter A.HORV ′ATHY Laboratoire de Math′e matiques et de Physique Th′e orique,Universit′e de Tours,Parc de Grandmont,F-37200Tours,France E-mail:horvathy@lmpt.univ-tours.fr Received September 25,2006,in final form November 27,2006;Published online December 14,2006Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper090/Abstract.Non-commutative structures were introduced,independently and around the same time,in mathematical and in condensed matter physics (see Table 1).Souriau’s construction applied to the two-parameter central extension of the planar Galilei group leads to the “exotic”particle,which has non-commuting position coordinates.A Berry-phase argument applied to the Bloch electron yields in turn a semiclassical model that has been used to explain the anomalous/spin/optical Hall effects.The non-commutative parameter is momentum-dependent in this case,and can take the form of a monopole in momentum space.Key words:non-commutative mechanics;semiclassical models;Hall effect 2000Mathematics Subject Classi?cation:81V70;81T751“Exotic”Galilean symmetry and mechanics in the plane Central extensions first entered physics when Heisenberg realized that,in the quantum me-chanics of a massive non-relativistic particle,the position and momentum operators did not commute.As a consequence,phase-space translations act only up-to-phase on the quantum Hilbert space.In more mathematical terms,it is not the [commutative]translation group it-self,only its [non-commutative]1-parameter central extension,the Heisenberg group,which is represented unitarily.Similarly,Galilean boosts act,for a massive non-relativistic system,only up-to phase.In other words,it is the 1-parameter central extension of the Galilei group,called the Bargmann group,that acts unitarily.True representations only arise for massless particles.Are there further extension parameters?In d ≥3space dimensions,the Galilei group admits a 1-parameter central extension only [1].The extension parameter,m ,is identified with the physical mass.However,L′e vy-Leblond [2]recognized that,in the plane,the Galilei group admits a second extension,highlighted by the non-commutativity of the Galilean boost generators,[K 1,K 2]=iκ,

where κis the new extension parameter.This has long been considered,however,a mere mathematical curiosity,as planar physics has itself been viewed a toy.The situation started to change around 1995,though,with the construction of physical models which realize this “exotic”symmetry [3,4,5,6,7,8].These models have the strange feature that the Poisson bracket of

2P.A.Horv′a thy m2

≡θ.

Table1.Exotic Galilean symmetry vs.semiclassical models with Berry term.

HIGH-ENERGY/MATH.PHYS.

1970L′e vy-Leblond:

1995–2000Niu et al.:Berry term mechanical models with exotic symmetry

2000–2001Duval et al.:

Anomalous Hall effect

non-commutative mechanics

2004B′e rard,Mohrbach:

momentum space in Anomalous Hall effect non-commutativity

spin-Hall effect

2005Sinova et al:

relativistic spin

2005Duval et al:“SpinOptics”

Bliokh:Optical Magnus/Hall effect

2The exotic model

In[3,4,5,6,9,10,11,12]Souriau’s“orbit method”[13]was used to construct a classical planar system associated with L′e vy-Leblond’s two-fold extended Galilean symmetry.It has an“exotic”symplectic form and a free Hamiltonian,

?0=dp i∧dx i+1

2m

.

The associated(free)motions follow the usual straight lines;the“exotic”structure behaves, roughly,as spin:it enters the(conserved)boost and the angular momentum, j=?ij x i p j+

θ

Non-Commutative Mechanics in Mathematical&in Condensed Matter Physics3

2dt+

θ

m?θ,{x i,p j}=

m

m?

eB.

A remarkable property is that for vanishing e?ective mass m?=0,i.e.,when the magnetic field takes the critical value B=1/(eθ),the system becomes singular.Then“Faddeev–Jackiw”(alias symplectic)reduction yields an essentially two-dimensional,simple system,reminiscent of “Chern–Simons mechanics”[17,18]1.The symplectic plane plays,simultaneously,the role of both configuration and phase space.

The clue is to introduce the“twisted”coordinate

Q=r?q,r=(x i),q i=?ij

p j

B

.

4P.A.Horv′a thy

eB ,H red=eV(Q1,Q2)+

θ2e2m

B

.

Exotic oscillator:E=?ω2r.The general motions follow elliptical orbits.In the critical case θB=1,however(see Fig.4),the guiding center and the real-space position become proportional, q=(1+θ2ω2)r.The only consistent motions are circular,with“Hall”angular velocity ?=

ω2B

2 1+ω2θ2 Q2∝I red=B

1+θ2ω2(1/2+n)

,n=0,1,....

Non-Commutative Mechanics in Mathematical &in Condensed Matter Physics 5

?p j

.

Its curvature,

Θ(p )=?p ×A l (p )is hence purely momentum-dependent.

Then the authors of

[23,

24]

argue that the semiclassical equations of motion in n th band should be modified by including the Berry term,according to

˙r =??n (p )

1+e B ·Θ,{x i ,p j }Bloch =δi +eB i Θj

1+e B ·Θ

6P.A.Horv′a thy p3

,(6) p=0.This is indeed the only possibility consistent with rotational symmetry[41,42].

For B=0and a constant electric field,E=const and assuming a parabolic profile?n(p)= p2/2,equation(4)with non-commutative parameter(6),˙p=e E,is integrated as p(t)= e E t+p0.The velocity relation(3)becomes in turn

˙r=p0+e E t+

eθEk0

2

eEt2.(Our choices correspond to choosing time so that the turning point is at t=0.)However,due to the anomalous term in(3),the particle is also deviated perpendicularly to p0and E,namely by

z(t)=

θ

p0

.(8)

Cf.Fig.5,then continue withθbecomes a half-integer upon quantization,θ=N/2,and hence(8)is indeed N/k0.The constant p0=0,the minimal possible value of momentum,plays the role of an impact parameter.Let us observe that while(8)does not depend on the field E or the electric charge e,the limit eE→0is singular.For eE=0,the motion is uniform along a straight line.

Non-Commutative Mechanics in Mathematical&in Condensed Matter Physics7

5The expression(9)of the total angular momentum is not mandatory,since,for a free particle,the two terms are separately conserved.

8P.A.Horv′a thy

Non-Commutative Mechanics in Mathematical&in Condensed Matter Physics9

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23. 饱和温度saturation temperature 24. 饱和压力saturation pressure 25. 饱和液体saturation liquid 26. 饱和蒸汽saturation vapor 27. 过饱和supersaturation 28. 湿蒸汽wet vapor 29. 过热superheat 30. 过热蒸汽superheated vapor 31. 过热度degree of superheat 32. 过冷subcooling 33. 过冷液体subcooled liquid 34. 气-液混合物liquid-vapor mixture 35. 干度quality 36. 冷凝点condensation point 37. 冷凝液condensate 38. 共沸混合物azeotropic mixture 39. 共沸性azeotropy 40. 共沸点azeotropic point 41. 凝固solidification 42. 熔化fusion 43. 熔点melting point 44. 升华sublimation

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