18 Gauge Fields, Coordinate Systems, and Geometric Phases

18 Gauge felds, coordinate systems, and geometric phases

18.1 Maxwell’s equations

Maxwell said

1 ∂B! Faraday’s law E! = (18.1) ∇× − c ∂t 1 ∂D! Ampère’s law H! = !j + (18.2) ∇× c ! ∂t " Coulomb’s law D! = ρ (18.3) ∇ no magnetic monopoles B! =0 (18.4) ∇ and it was light [43, 44]. Together with Newton’s laws of classical mechanics and the Lorentz force !v F! = q E! + B! (18.5) c × # $ that was all what was known fundamentally about physics by the end of the 19th century. Maxwell originally didn’t write his equations in the modern notation using vectors we are familiar with. This was introduced later by Oliver Heaviside [45] and Josiah Willard Gibbs, making the equations much easier to read (and evaluate). Instead, Maxwell used 20 scalar felds, inspired by the then-popular fuid mechanics. Also his equations contain explicitly the vector potential A!, which can be eliminated from his equations. The equations for the 16 parameters above above include the relations

B! = µ0 H! + λM! , (18.6) % & D! = %0E! + λP,! (18.7) introducing the macroscopic magnetization M! and electric polarization P! with the magnetic permeability µ0 and the dielectric constant %0. The factor λ denotes a dimension-

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Table 18.1: Defnitions of µ0, %0, and λ in different unit systems

Unit system µ0 %0 λ 7 2 Système Internationale 4π 10− T /(A/m) 1/(µ0c )1 × Gauß 1 14π Heaviside-Lorentz 1 11 esu 1/c2 14π emu 1 1/c2 4π less constant depending on the unit system chosen.1 We are using Heaviside-Lorentz units (named after Oliver Heaviside and Hendrik Antoon Lorentz), equivalent to setting µ0 = %0 = λ =1. This makes the structure of Maxwell’s equations particularly easy to understand.2

18.1.1 The scalar and vector potentials In the following, we set the magnetization M! =0and the polarization P! =0. That is, we express felds E! and B! in terms of all charges and currents on a microscopic level. In this form, Maxwell’s equations are often named «in vacuum» (which is wrong or at least misleading) or «microscopic» in contrast to the «macroscopic» equations above involving M! and P! .

1Setting λ =4π (Gauß, esu, emu) scales electromagnetic quantities by an integration over the solid angle. The factor 4π then appears explicitly in Coulomb’s law (18.3), however drops out of calculations involving planar geometries, for example parallel charged plates. 2Maxwell’s equations are equations each establishing a certain proportionality. Historically, different proportionality constants have emerged, leading to different electromagnetic unit systems. Among these are the Système Internationale d’Unités, the Gaussian unit system, and the Heaviside-Lorentz system, see Table 18.1 for the corresponding values of the constants. Using these constants, we can express the Coulomb force between two point-like charged particles as

λ q1q2 F = 2 (18.8) 4π#0 r

and the force between two infnitely thin wires of length $ at distance r as

λµ0 $ F = 2I1I2 . (18.9) 4π r Maxwell’s equations then have the form

1 1 ∂B% 1 1 ∂D% √#0 E% = , √µ0 H% = λ%j + , D% = λρ, B% =0. ∇× − √µ0 c ∂t ∇× √#0 c ! ∂t " ∇ ∇

Everything else one needs to know about electromagnetic units can be found in the appendix of John David Jackson’s book on electromagnetism.

282 18.1 Maxwell’s equations

With Stokes’ theorem in differential form,

( A!)=0, (18.10) ∇· ∇× ( φ)=0 (18.11) ∇× ∇· for any suffciently well-behaved vector feld A! = A!(!x, t) and scalar feld φ = φ(!x, t),we can use the divergence-freeness of the magnetic feld (18.4) and Faraday’s law (18.1)to express E! and B! through

B! = A,! (18.12) ∇× 1 ∂ E! + A! =0 (18.13) ∇× c ∂t # $ 1 ∂ E! = A! + φ . (18.14) ⇒ − c ∂t ∇ # $ We replace E! and B! accordingly in Ampère’s law (18.2) and obtain 1 ∂ 1 ∂ 1 A! φ A! = !j. (18.15) ∇× ∇× − c ∂t −∇ − c ∂t c % & # $ Using the identity A! = A! ( ) A! (18.16) ∇× ∇× ∇ ∇ − ∇·∇ % & % & ∆ and the d’Alembert operator ' () * 1 ∂2 := ∆ (18.17) ! c2 ∂t2 − we obtain 1 ∂ 1 A! + φ + A! = !j. (18.18) ! ∇ c ∂t ∇ c # $ Replacing E! from Eq. (18.14) in Coulomb’s law (18.3) yields 1 ∂ φ A! = ρ (18.19) ∇ −∇ − c ∂t # $ 1 ∂ ∆φ A! = ρ (18.20) ⇒− − c ∂t∇

1 ∂ 1 ∂ φ φ + A! = ρ, (18.21) ⇒ ! − c ∂t c ∂t ∇ # $ where in the last line we just added and subtracted (1/c2)∂2φ/∂t2. We note that the equations (18.18) and (18.21) are exactly equivalent to Maxwell’s equations: Four frst- order differential equations are replaced by two second-order equations.

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18.2 Gauge invariance

The felds A! and φ are not uniquely determined by these equations: Shifting with Λ= Λ(!x, t) according to

A! A! + Λ=: A!", (18.22) → ∇ 1 ∂ φ φ Λ=: φ", (18.23) → − c ∂t doesn’t change the physical felds B! and E! :

B! = A! A!" = A! + Λ ∇× →∇× ∇× ∇ (18.24) = A! = B,! % & ∇× 1 ∂ 1 ∂ 1 ∂ 1 ∂ E! = φ A! φ" A!" = φ Λ A! + Λ −∇ − c ∂t → −∇ − c ∂t −∇ − c ∂t − c ∂t ∇ # $ (18.25) 1 ∂ % & = φ A! = E.! −∇ − c ∂t This is called a gauge degree of freedom: The felds A! and φ are defned only up to a gradient of a (possibly space- and time-dependent) scalar feld Λ, and any particular choice of Λ (called «fxing the gauge») leads to exactly the same physics. Therefore any theory involving electromagnetic felds must be formulated in a form which respects this gauge invariance. Including the Lorentz force into Lagrange’s equations ∂ d ∂ L L =0 (18.26) ∂xi − dt ∂x˙ i where ∂ /∂xi are the components of the canonical force and pi = ∂ /∂x˙ i are the com- L L ponents of the canonical momentum leads to e p! = m!x˙ + A! (18.27) c for the canonical momentum of a particle with charge e. We stress that the canonical momentum p! , being just a variable conjugate to the position !x explicitly depends on the gauge chosen. Thus in order to write the kinetic momentum in the Lagrange function in a gauge invariant form, we need to substitute it according to e m!x˙ p! A,! (18.28) → − c a substitution we will meet again in quantum mechanics, where in real-space representation p! =(h¯/i) is the operator entering the canonical commutation relation [!x,p ! ]= ∇ i¯h.

284 18.2 Gauge invariance

18.2.1 Common gauges The Lorenz gauge This is named after the danish physicist Ludvig Valentin Lorenz, not to be confused with Hendrik Lorentz mentioned above. It reads 1 ∂ φ + A! =0, (18.29) c ∂t ∇ turning Maxwell’s equations into a set of inhomogeneous wave equations, particularly popular in the calculation of time dependent electromagnetic felds. This does not fx the gauge completely, as any scalar ψ = ψ(!x, t) satisfying a homogeneous wave equation

! ψ =0 (18.30) can still be used to shift A! A! + ψ and φ φ (1/c)∂ψ/∂t without changing the → ∇ → − physics. We note that the Lorenz gauge is Lorentz invariant.

The Coulomb gauge Here we set A! =0. (18.31) ∇ Eq. (18.21) then reduces to Poisson’s equation,

∆φ = ρ, (18.32) − the solution of which is the instantaneous Coulomb potential.3 This also allows us to write the vector potential for a static uniform magnetic feld (a very common experi- mental setup in solid-state physics) as 1 A! = !x B.! (18.33) −2 ×

Two remarks: (a) The Coulomb gauge is not Lorentz invariant. (b) A! diverges.

The temporal gauge In this gauge, the scalar feld itself φ =0. (18.34)

Used by Hermann Weyl who discovered the principle of gauge invariance in quantum mechanics in 1918, quite some years before Schrödinger’s equation was established.

3A contradiction to Einstein’s postulate of a fnite speed of light c, refecting the fact that the scalar potential itself is not a physical object.

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The axial gauge A very special choice,

A3 =0. (18.35)

Used in particle physics to avoid «ghost felds».

18.3 Nonrelativistic quantum mechanics

In this section we discuss the Hamiltonian for a single nonrelativistic scalar particle of mass m, charge e. For vanishing electromagnetic felds, its Hamiltonian and the Schrödinger equation of motion simply reads

2 1 h¯ H = , (18.36) 2m i ∇ # $ ∂ HΨ=i¯h Ψ. (18.37) ∂t A fundamental principle of quantum mechanics states that the properties of a physical object cannot be determined by measuring its wavefunction (which is impossible), but rather only by determining expectation values of operators expressed in a certain representation.4 That is equivalent to the freedom to choose an arbitrary phase for the wavefunction: This phase cannot be expressed as the expectation value of an operator, thus cannot be measured, hence its choice is arbitrary and doesn’t affect the physics of the system. So if we set ie/(hc¯ )Λ Ψ ΨΛ := e Ψ, (18.38) → Schrödinger’s equation should not be changed. However taking the gradient of the shifted wavefunction shifts the momentum operator and the time derivative

h¯ h¯ e + Λ, (18.39) i ∇→ i ∇ c∇ ∂ ∂ e ∂ i¯h i¯h Λ (18.40) ∂t → ∂t − c ∂t and therefore does change the form of Eq. (18.37) except for the trivial case of a constant, single-valued global phase. But we require that Schrödinger’s equation is invariant under local phase transformations Λ=Λ(!x, t). We take this into account by modifying the Hamiltonian such that 2 1 h¯ e H = A! + eφ (18.41) 2m i ∇−c # $ 4For the purpose of this section, we choose real-space representation with p% =(h¯/i) for the canonical ∇ momentum in the absence of electromagnetic felds.

286 18.4 Coordinate systems and require that under the gauge transformation

1 ∂ A! A! + Λ,φ φ Λ (18.42) → ∇ → − c ∂t the physics described by the Hamiltonian doesn’t change. This is called minimal cou- pling, as it is the most simple way of achieving the gauge invariance sought. Transform- ing the phase of the wavefunction according to (18.38) now leads back to the original Schrödinger equation. A little gymn shows

2 1 h¯ e 1 ∂ ∂ A! + Λ + e φ Λ eie/(hc¯ )ΛΨ=i¯h eie/(hc¯ )ΛΨ (18.43) +2m i ∇−c ∇ − c ∂t . ∂t , % &- # $ 2 1 h¯ e 1 ∂ ∂ e ∂ eie/(hc¯ )Λ A! + e φ Λ Ψ=eie/(hc¯ )Λ ih¯ Λ Ψ ⇒ 2m i ∇−c − c ∂t ∂t − c ∂t / # $ # $0 # $ 2 1 h¯ e ∂ A! + eφ Ψ=ih¯ Ψ. (18.44) ⇒ 2m i ∇−c ∂t / # $ 0

A phase transformation according to Eq. (18.38) is a one-dimensional unitary transformation: We have extended Schrödinger’s equation such that it is invariant under U(1) gauge transformations, and A!(!x, t) together with φ(!x, t) is called a gauge feld.5 This concept is generalized in quantum feld theory to construct equations of motion for the particles in the standard model, which is comprised of the electroweak interaction for the three lepton families (gauge theory with U(1) SU(2) gauge symmetry) and the quan- × tum chromodynamics for the three quark families (SU(3) gauge symmetry). The gauge felds in these theories correspond to the exchange bosons, which are the photon and + 6 the three intermediate vector bosons W ,W−, and Z together with the eight gluons.

18.4 Coordinate systems

According to Terence Tao [46], a gauge transformation like Eq. (18.42) can be regarded as a somewhat complicated coordinate transformation. So what is a coordinate system?

5Eq. (18.41) implicitly contains that in a classical theory magnetism cannot exist, also known as the Bohr- van-Leeuwen theorem of statistical mechanics. The essence of the theorem is that if the magnetic feld exclusively enters the canonical momentum p% (e/c)A%, we can always choose the origin of the phase- − space integration of the free energy F such that A% is eliminated and therefore the magnetization M% = (1/V )∂F/∂B% must vanish identically. So any kind of magnetism is a purely quantum-mechanical − phenomenon. 6There is one additional particle, the Higgs boson, however that’s the only boson in the standard model which is not a gauge particle.

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From linear algebra we have learned that for a family =(v1,...,vn) of vectors of a B K-vector space V exists exactly one linear mapping

n Φ : K V with Φ (x1,...,xn)=(x1v1,...,xnvn). (18.45) B → B

If is a basis of V , Φ is an isomorphism. In this case Φ is called a coordinate system B B B of V , and for one v V , the vector ∈

1 x =(x1,...,xn) := Φ− (v) (18.46) B is called the coordinate vector of v with respect to . So associating V with the family of B gauge felds (one four-component feld in case of electromagnetism) indeed means that «fxing a gauge» technically can be regarded as choosing a particular coordinate system, i. e., choosing a particular representation of (φ(!x, t), A!(!x, t)). However I don’t see where this might help solving Maxwell’s equations, apart from the general concept that it is always wise to choose a representation of the mathematical objects one is dealing with such that expressing the problem to be solved in terms of the chosen representation makes the solution easier to achieve.

18.5 Geometric phases

We have pointed out that in general gauge felds like the vector potential A! and the scalar potential φ don’t have a physical meaning. For classical electrodynamics that might be true, however in quantum mechanics on non-Euclidean geometries and nontrivial topologies, they actually do have an infuence: In general a wave function of a system considered can acquire an additional phase factor when passing a closed path in some parameter space. This is known as geometric or Berry phase, named after Michael Berry who made the general concept popular in 1983.

18.5.1 A classical example: Foucault’s pendulum

Geometric phases appear in classical mechanics as well, and we have an example in the institute: Foucault’s pendulum at the entrance. According to Léon Foucault, the plane of oscillation of the pendulum rotates with an angular velocity ω = ω sin θ where ω E E ≈ 360◦/24 h is the angular velocity of the earth and θ the latitude of the point p where the pendulum is hanging. For our institute, my iPhone tells me that θ 51◦1"36"". Thus after ≈ one day when p returns to its original position, the plane of oscillation of our pendulum has rotated by only φ 280◦, which is the geometric phase it has acquired. (It takes our ≈ pendulum 30:47:11 hours to complete a full rotation.)

288 18.5 Geometric phases

Figure 18.1: Abstract by Walter Franz on electron interferences in a magnetic feld, 1939.

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18.5.2 Aharonov-Bohm effect A well-known quantum mechanical example is the Aharonov-Bohm effect published in 1959, discussing the interference pattern of electron beams in a magnetic feld. A small literature search however reveals that already 20 years earlier, Walter Franz from Königsberg (Preußen), today Kaliningrad, submitted a remarkable contribution entitled Elektroneninterferenzen im Magnetfeld (electron interferences in a magnetic feld) to the Verhandlungen der Deutschen Physikalischen Gesellschaft, which is an abstract [47] for the Tagung des Gauvereins Ostland (conference of the Gauverein Ostland) held in May 1939 in the (then for a few months still) Free City of Danzig, today Gdansk.´ Fig. 18.1 shows the original text with a discussion of the impact of a vector potential A on the interference pattern of the electrons in the double-slit experiment, relating the de Broglie wavelength λ = h/ p to the canonical momentum p = mv +(e/c)A. His line of ar- | | guments is exactly the same as ours in the example that follows, the only difference being that we avoid the picture of korpuskularen Bahnen (particle trajectories) for the electrons.

18.5.3 Persistent currents in small metallic rings Another manifestation of a geometric phase is a persistent current in a small metallic ring R with circumference L enclosing a fux φ, which we shall discuss here in detail. Fig. 18.2 shows the geometry. According to Eq. (18.41), the Hamiltonian for N particles with charges ej, j =1,...,N in the ring reads

H = H p! j (ej/c)A!(!r j),!rj ,j=1...N, (18.47) − % & with a magnetic feld B! = A!, fux φ = A!d!s through the ring. The magnetic feld ∇× ∂S supposedly vanishes on the ring, B! (!r )=0for !r R, therefore on the ring we have 1 ∈ A! =0 (18.48) ∇× A! = χ(!r ),!rR (18.49) ⇒ ∇ ∈ φ = χd!s ⇒ ∇ 2∂S = χ(L) χ(0); (18.50) − H = H (p! j (ej/c) jχ(!r j),!rj) . (18.51) − ∇ Ansatz:

i ej Ψ(!r j)=Ψ0(!r j) exp χ(!r j) (18.52) h¯ c  5j HΨ(!r j)=H0Ψ0(!r j)=EΨ(!r j),H0:= H(p! j,!rj). (18.53) ⇒

290 18.5 Geometric phases

Figure 18.2: A small metallic ring R of circumference L enclosing a magnetic fux φ.

Assume ej = ne, n Z j =1,...,N. Then ∈ ∀

ie ˜ ˜ Ψ(!r j)=Ψ0(!r j) exp Nχ(!r j) , N Z, (18.54) hc¯ ∈ , - specifcally in one dimension (coordinate x along the ring)

ieN˜χ(x)/(hc¯ ) Ψ(x)=Ψ0(x)e

2πiN˜χ(x)/φ0 hc Ψ0(x)e ,φ0 := ≡ e 2πiχ(x)/φ0 =Ψ0(x)e . (18.55)

Uniqueness of the wavefunction and its derivative:

Ψ(L) Ψ0(L) 2πi(χ(L) χ(0))/φ0 ! = e − =1 (18.56) Ψ(0) Ψ0(0) Ψ(L)=Ψ(0)e2πiφ/φ0 , (18.57) ⇒ analogously dΨ dΨ = e2πiφ/φ0 . (18.58) x x d 8L d 80 8 8 8 8 In analogy to Bloch electrons 8 8

! i%kR% Ψ%k(!r + R)=e Ψ%k(!r ) (18.59) we identify the phases according to

2πφ L ˆ= R,! ˆ= !k. (18.60) Lφ0

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For the current of the nth energy level En at T =0we then get

e 1 ∂En In = vn,vn = , (18.61) −L h¯ ∂k or using the analogy above

Lφ0 ∂En Lc ∂En vn = = (18.62) 2πh¯ ∂φ e ∂φ ∂En In = c (18.63) ⇒ − ∂φ for a single level. The resulting net current is

N ∂En IN = c , (18.64) − ∂φ n=1 5 or for fnite temperatures

N ∂En ∂F IN = c f(En) = c (18.65) − ∂φ − ∂φ n=1 5 with the free energy F and the Fermi-Dirac distribution f(x).

For free electrons we have intersecting parabolas for En(φ); if there are defects, the degeneracies at the intersections are lifted (like with nearly free electrons on a lattice) and the current is correspondingly smaller. Because En(φ) is symmetric in φ, the current In is antisymmetric and vanishes at φ =(,/2)φ0, , Z. ∈ Spinless free electrons:

2 h¯2 2π φ E = n + n 2m L φ , # 0 $- 2 (2πh¯)2 φ = n + , (18.66) 2mL2 φ # 0 $ 2πh¯ φ In = n + , (18.67) −mL2 φ # 0 $ N 2 N even n =0, 1, 2,..., ±N 1 . (18.68) ± ± + ± N odd ± 2 At the Fermi level, the tiny level spacing at φ =0is given by

2πhv¯ ∆= F (18.69) L

292 18.5 Geometric phases with two states per interval ∆.AtT =0we obtain the total current by summing up all contributions from levels with energy less than µ. For a single ring with a fxed number of electrons (µ = h¯2(Nπ)2/(2mL2)), we get

2φ 1 φ 1 I0 N odd − φ0 − 2 ≤ φ0 ≤ 2 IN (φ)= 2φ φ , (18.70) + I0 1 N even 0 1 − φ0 − ≤ φ0 ≤ πehN¯ % evF & I0 = = . (18.71) mL2 L For a general chemical potential µ, N varies between even and odd as a function of φ, therefore also the resulting current. We note that although there is no magnetic feld whatsoever on the ring (and although the ring is not a superconductor), a small persistent current fows through the ring due to the presence of fux. Problem: fnite temperatures. Because at fnite T , we have f(En) f(En+1) also for ≈ levels near the Fermi energy, even and odd contributions to IN almost exactly cancel. Additionally the electrons lose their phase coherence due to impurity scattering and interaction with phonons, such that the measurable current should be very small.

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