130B Quantum Physics

Part 5. Phase and Gauge

Gauge Principles

Gauge Structure and Berry Phase

Phase Ambiguities

p(x)=((|ψ(x)|)^2),

Global phase ambiguity. A global phase rotation of the wavefunction (where “global” means α does not depend on x)

has no consequence on the expectation value ⟨O⟩ of any physical observable O in any case

⟨O⟩=∫(SuperStar[ψ])(x)O(x,x')ψ(x')(d^D)x(d^D)x'.

Conclusion: quantum states ∈ projective Hilbert space, where global phase is always unphysical.

Local phase ambiguity (Gauge redundancy). We can push this idea further: if we restrict ourself to diagonal observables in the position basis, i.e., functions f(x) that depends only on x (but not p), then any local phase rotation

will leave all expectation values ⟨f(x)⟩ invariant,

⟨f(x)⟩:=∫f(x)((|ψ(x)|)^2)(d^D)x=∫f(x)p(x)(d^D)x

since the probability density p(x) is unchanged.

If p(x)=((|ψ(x)|)^2) was the only physical probability distribution to be modeled, any ψ(x) related by local phase rotation Eq. () should be treated as equivalent.

This represents a gauge redundancy: multiple mathematical descriptions (e.g. wavefunctions) describing the same physical reality (e.g. position distribution).

🤔 Quantum decoherence provides a deeper reason of why only diagonal observables are measurable.

Environmental monitoring: The environment has a natural tendency to monitor the local density p(x) of particles in the space.

Effectively performing weak continuous measurement of x.

Inducing decoherence in the position basis: rapid decay of off-diagonal coherence ρ(x,x') of the density matrix for x≠x'.

Consequence 1: Dephasing noise.

The measurement randomizes the phase of ψ(x) at every position independently.

Relative phase between ψ(x) and ψ(x') no longer comparable, allowing local phase rotation as Eq. () to become a redundancy.

Consequence 2: Gauge projection.

The measurement collapse the system towards the particle number eigenstates, suppressing the number fluctuations.

Effectively imposing a gauge constraint (like Gauss law), that couples the particle to an emergent gauge field, allowing particles to interact with each other through emergent gauge forces.

Gauge Transformation

ψ(x,t)→(e^(i χ(x,t)))ψ(x,t),

Start from the free Schrödinger equation

i ℏ(∂_t)ψ(x,t)=-((ℏ^2)/(2m))(∇^2)ψ(x,t),

under gauge transformation ψ→(e^(i χ))ψ in Eq. (), the derivative operators picks up extra terms involving (∂_t)χ and ∇χ,

i ℏ ((∂_t)+i (∂_t)χ(x,t))ψ(x,t)=-((ℏ^2)/(2m))(((∇+i ∇χ(x,t)))^2)ψ(x,t),

and the equation does not remain invariant.

Show that Eq. () becomes to Eq. () under gauge transformation.

({{(∂_t)ψ, →, (∂_t)((e^(i χ))ψ)}, { , =, (∂_t)(e^(i χ))ψ+(e^(i χ))(∂_t)ψ}, { , =, i (∂_t)χ (e^(i χ))ψ+(e^(i χ))(∂_t)ψ}, { , =, (e^(i χ))((∂_t)+i (∂_t)χ)ψ,}})

({{∇ψ, →, ∇((e^(i χ))ψ)}, { , =, ∇(e^(i χ))ψ+(e^(i χ))∇ψ}, { , =, i ∇χ (e^(i χ))ψ+(e^(i χ))∇ψ}, { , =, (e^(i χ))(∇+i ∇χ)ψ.}})

(e^(-i χ))(∂_t)(e^(i χ))=(∂_t)+i (∂_t)χ,
(e^(-i χ))∇(e^(i χ))=∇+i ∇χ.

i ℏ(∂_t)ψ=-((ℏ^2)/(2m))(∇^2)ψ.

Under gauge transformation, replace ψ→(e^(i χ))ψ,

i ℏ(∂_t)((e^(i χ))ψ)=-((ℏ^2)/(2m))(∇^2)((e^(i χ))ψ).

Commute (e^(i χ)) through the derivative operators

i ℏ (e^(i χ)) ((∂_t)+i (∂_t)χ)ψ=-((ℏ^2)/(2m))(((e^(i χ))(∇+i ∇χ))^2)ψ.

Cancel (e^(i χ)) from both sides, we arrive at the Schrödinger equation after the transform

i ℏ ((∂_t)+i (∂_t)χ)ψ=-((ℏ^2)/(2m))(((∇+i ∇χ))^2)ψ.

To restore the gauge invariance, we introduce gauge fields:

Scalar potential: Φ(x,t) - a scalar field in the spacetime

Vector potential: A(x,t) - a vector field in the spacetime

and replace derivatives by covariant derivatives:

i ℏ(∂_t)ψ(x,t)=((1/(2m))(((-i ℏ∇-A(x,t)))^2)+Φ(x,t))ψ(x,t).

Semiclassical Interpretation

ψ=A (e^(i S/ℏ)).

(-(∂_t)S-Φ)=(1/(2m))(((∇S-A))^2).

Scalar potential Φ: potential energy,

Vector potential A: potential momentum.

({{ , Total, =, Kinetic, +, Potential}, {Energy, E, =, (1/2)m((OverDot[x])^2), +, Φ}, {Momentum, p, =, m(OverDot[x]), +, A}})

Show that both equations in Eq. () are consistent with Eq. ().

p=m(OverDot[x])+A,

E=(1/(2m))(((p-A))^2)+Φ
=(1/(2m))(((m(OverDot[x])))^2)+Φ
=(1/2)m((OverDot[x])^2)+Φ.

Total (or Canonical): appear directly in conservation laws and determine the action accumulated in spacetime.

Kinetic: directly linked to the particle’s motion (velocity (OverDot[x])).

Potential: exist independently of particle motion, contributing even when the particle is at rest ((OverDot[x])=0), representing the interaction with the background field in the spacetime.

F=(d/(dt))(m (OverDot[x]))=((dp)/(dt))-((dA)/(dt)),

dp/dt=(∂_t)p+(OverDot[x])·∇p, in which

∇p=0, as x and p are independent variables,

Maxwell relation: (∂_t)∇S=∇(∂_t)S implies

(∂_t)p=-∇E=(∇A)·(OverDot[x])-∇Φ.

dA/dt=(∂_t)A+(OverDot[x])·∇A.

Justify Eq. () and Eq. ().

Eq. ():

(∂_t)p=-∇E
=-∇((1/2)m((OverDot[x])^2)+Φ)
=-∇((1/2)m((OverDot[x])^2))-∇Φ
=-m(∇(OverDot[x]))·(OverDot[x])-∇Φ
=-(∇(m(OverDot[x])))·(OverDot[x])-∇Φ
=-(∇(p-A))·(OverDot[x])-∇Φ
=(-∇p+∇A)·(OverDot[x])-∇Φ
=(∇A)·(OverDot[x])-∇Φ.

Eq. (): the first term E=-∇Φ-(∂_t)A is a simple substitution, so we focus on the second term (OverDot[x])×B and show that

(Subscript[((OverDot[x])×B), i])=(Subscript[((OverDot[x])×(∇×A)), i])
=(ϵ^(i j k))(Subscript[(OverDot[x]), j])(ϵ^(k l m))(∂_l)(Subscript[A, m])
=((δ^(i l))(δ^(j m))-(δ^(i m))(δ^(j l)))(Subscript[(OverDot[x]), j])(∂_l)(Subscript[A, m])
=(Subscript[(OverDot[x]), j])(∂_i)(Subscript[A, j])-(Subscript[(OverDot[x]), j])(∂_j)(Subscript[A, i])
=(Subscript[((∇A)·(OverDot[x])-(OverDot[x])·∇A), i]).

Math Interlude: Lorentz Vectors

Spacetime: Introduce x=((x^0),(x^1),(x^2),(x^3)) to denote the coordinate of a spacetime point,

t=(x^0): time,

x=((x^1),(x^2),(x^3)): space,

and denote these components as (x^μ)(μ=0,1,2,3) jointly. x is said to be a Lorentz vector.

Spacetime derivatives: Partial derivatives in the spacetime are defined as

(∂_μ)f(x)=(lim_(δ,x->0))((f(x+δ,x)-f(x))/(δ,(x^μ))),

(∂_t)=(∂_0): temporal derivative,

∇=((∂_1),(∂_2),(∂_3)): spatial derivatives.

Energy-momentum: Introduce p=((p^0),(p^1),(p^2),(p^3)) to denote the energy and momentum,

E=(p^0)=-(Subscript[p, 0]): energy,

p=((p^1),(p^2),(p^3))=((Subscript[p, 1]),(Subscript[p, 2]),(Subscript[p, 3])): momentum.

Gauge field: Introduce A=((A^0),(A^1),(A^2),(A^3)) to denote the gauge field,

Φ=(A^0)=-(Subscript[A, 0]): scalar potential,

A=((A^1),(A^2),(A^3))=((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[A, 3])): vector potential.

Covariant derivatives: Operators in Eq. () can now be unified as a single Lorentz vector operator

(Subscript[D, μ])=(∂_μ)-(i/ℏ)(Subscript[A, μ]),

(Subscript[D, t])=(Subscript[D, 0]): covariant temporal derivative,

D=((Subscript[D, 1]),(Subscript[D, 2]),(Subscript[D, 3])): covariant spatial derivative.

({{(a^μ)(Subscript[b, μ]), ✓, ok}, {(a^μ)(b^μ), ×, no!}, {(Subscript[a, μ])(Subscript[b, μ]), ×, no!}})

(a^μ)(Subscript[b, μ])=(a^0)(Subscript[b, 0])+(a^1)(Subscript[b, 1])+(a^2)(Subscript[b, 2])+(a^3)(Subscript[b, 3])
=(a^μ)(Subscript[g, μ ν])(b^ν)=-(a^0)(b^0)+(a^1)(b^1)+(a^2)(b^2)+(a^3)(b^3)
=(Subscript[a, μ])(g^(μ ν))(Subscript[b, ν])=-(Subscript[a, 0])(Subscript[b, 0])+(Subscript[a, 1])(Subscript[b, 1])+(Subscript[a, 2])(Subscript[b, 2])+(Subscript[a, 3])(Subscript[b, 3]).

Berry Phase

A processes is said to be adiabatic, if it happens slowly over a long time, i.e. the rates of change in physical observables tend to zero. In terms of the motion of a particle, it means the velocity of the particle is almost zero throughout the process:

(OverDot[x])->0.

Eq. () is also called the adiabatic limit.

In the adiabatic limit, the action of the particle is accumulated by the potential energy and momentum

-(∂_t)S=E=Φ,
∇S=p=A.

as the kinetic energy and momentum vanishes when (OverDot[x])->0.

Given that phase is related to action, the Berry phase that a particle accumulates along a spacetime trajectory 𝒞 is given by the path integral that computes the accumulated action

(Subscript[Θ, 𝒞])=((Subscript[S, 𝒞])/ℏ)=(1/ℏ)(∫_𝒞)-Φdt+A·dx=(1/ℏ)(∫_𝒞)(Subscript[A, μ])d(x^μ).

Justify Eq. ().

(Subscript[S, 𝒞])=(∫_𝒞)dS=(∫_𝒞)(∂_t)Sdt+∇S·dx
=(∫_𝒞)-Φdt+A·dx.

Graphics[{{AbsoluteThickness[1],Arrowheads[0.06],Arrow[{{0,0},{1,0}}],Arrow[{{0,0},{0,1}}]},{Text["\!\( x\)",{1,0},{-1.5,0}],Text["\!\( t\)",{0,1},{-0.2,-0.5}]},
GraphicsComplex[{{0.2,0.18},{0.3,0.4},{0.62,0.52},{0.8,0.64},{0.84,0.75},{0.8,0.86},{0.64,0.94}},{{Arrowheads[{{0.08,0.6}}],Arrow[BSplineCurve[{1,2,3,4,5,6,7}]]},{Point[{1,7}]},{Text["\!\(ψ\)",1,{1.8,-0.5}],Text["\!\(E\^\(I Θ\_𝒞\)ψ\)",7,{1,-0.5}]}}],Text["𝒞",{0.6,0.45}]},ImageSize->140]

For infinitesimal transportation 𝒞:x→x+δ,x,

Under gauge transformation

(Subscript[A, μ])(x)→(Subscript[A, μ])(x)+ℏ(∂_μ)χ(x),

the Berry phase along an open trajectory 𝒞 is not gauge invariant (hence not a physical observable):

(Subscript[Θ, 𝒞])->(Subscript[Θ, 𝒞])+χ((Subscript[x, end]))-χ((Subscript[x, start])),

where (Subscript[x, start]),(Subscript[x, end])=∂𝒞 are the starting and ending point of 𝒞.

the Berry phase around a closed loop (Wilson loop) is gauge invariant, and is a physical observable.

Gauge Field and Electromagnetism

Gauge Connection

(∂_x)f(x)=(lim_(δ,x->0))((f(x+δ,x)-f(x))/(δ,x)).

Problem: If the wavefunction ψ(x) has local phase ambiguity,

ψ(x)→(e^(i χ(x)))ψ(x),

meaning that ψ(x) at each point x is defined up to a phase rotation, there will be no basis to compare wavefunctions between distinct points.

☞ There is a similar issue in finance: you cannot directly compare currencies from different countries by their face values!

Are the following amounts of money the same?

Row[{Spacer[20],Graphics[{Inset[#1,{#2,0}]&@@@{{,-1},{,0},{,1}},Translate[{Text[#1,#2]&@@@{{"¥100",{-1,0}},{"$100",{0,0}},{"€100",{1,0}}}},{0,-0.3}]},ImageSize->180]}]

You should first move (parallel transport) the money to the same place before comparing.

Row[{Spacer[20],Graphics[{Inset[#1,{#2,0}]&@@@{{,-1},{,0},{,1}},Translate[{Text[#1,#2]&@@@{{"$13.3",{0,-0.4}},{"$100",{0,-0.2}},{"$108",{0,0}}},GrayLevel[0.7],Arrowheads[0.06],Arrow[{#,-0.4}&/@{-0.7,-0.3}],Arrow[{#,0}&/@{0.7,0.3}],Text[#1,#2]&@@@{{"¥100",{-1,-0.4}},{"€100",{1,0}}}},{0,-0.3}],Text[Style["03/27/2025",10,LightGray,"ConcreteText"],{1,-0.8},{0.3,-1}]},ImageSize->180]}]

During the conversion, the money will be multiplied by the exchange rate (exponential gauge connection), e.g.

(e^(0.0769610411361284`))€,100=$,108, (e^(-2.017406150760383`))¥,100=$,13.3.

Solution: Similarly, to define a meaningful derivative for the wavefunction, we have to introduce a gauge connection (Subscript[A, μ])(x) to keep track of the phase rotation needed to transport ψ(x+δ,x) back to the point x for comparison, for every point x along any direction μ.

(Subscript[D, μ])ψ(x)=(lim_(δ,x->0))(((e^(-(i/ℏ)(Subscript[A, ν])(x)δ,(x^ν)))ψ(x+δ,x)-ψ(x))/(δ,(x^μ))).

Interpretation: ψ(x+δ,x) has accumulated a Berry phase of exp((i/ℏ)(Subscript[A, ν])(x)δ,(x^ν)) compare to ψ(x) as the particle travels adiabatically. So when pulling ψ(x+δ,x) back to the point x, this phase should be compensated by the opposite phase factor exp(-(i/ℏ)(Subscript[A, ν])(x)δ,(x^ν)) before comparing.

Expressed in terms of the usual partial derivative modified by the gauge connection,

which exactly reproduces Eq. ().

Show Eq. () follows from Eq. () by taking the limit.

(Subscript[D, μ])ψ(x)=(lim_(δ,(x^μ)->0))(((e^(-(i/ℏ)(Subscript[A, ν])(x)δ,(x^ν)))ψ(x+δ,x)-ψ(x))/(δ,(x^μ)))
=(lim_(δ,(x^μ)->0))(((1-(i/ℏ)(Subscript[A, ν])(x)δ,(x^ν)+𝒪(δ,(x^2)))(ψ(x)+(∂_ν)ψ(x)δ,(x^ν)+𝒪(δ,(x^2)))-ψ(x))/(δ,(x^μ)))
=(lim_(δ,(x^μ)->0))((ψ(x)+(∂_ν)ψ(x)δ,(x^ν)-(i/ℏ)(Subscript[A, ν])(x)δ,(x^ν)ψ(x)+𝒪(δ,(x^2))-ψ(x))/(δ,(x^μ)))
=(lim_(δ,(x^μ)->0))((∂_ν)ψ(x)-(i/ℏ)(Subscript[A, ν])(x)ψ(x))((δ,(x^ν))/(δ,(x^μ)))+𝒪(δ,x)
=((∂_ν)ψ(x)-(i/ℏ)(Subscript[A, ν])(x)ψ(x))(Subscript[δ, μ]^ν)
=((∂_μ)-(i/ℏ)(Subscript[A, μ])(x))ψ(x).

The covariant derivative commute with the gauge transformation

as adapted from Eq. ().

The apparent deformation of the wavefunction ψ from x to x+δ,x can be expressed as

ψ(x+δ,x)=(e^((i/ℏ)(Subscript[A, μ])(x)δ,(x^μ)))(e^(δ,(x^μ)(Subscript[D, μ])))ψ(x),

which contains two contributions

the intrinsic deformation under parallel transport ψ→(e^(δ,(x^μ)(Subscript[D, μ])))ψ, which is generated by the covariant derivative (Subscript[D, μ]),

the background deformation in terms of the Berry phase ψ→(e^((i/ℏ)(Subscript[A, μ])(x)δ,(x^μ)))ψ accumulated along the gauge connection (Subscript[A, μ]).

Gauge Curvature

Row[{Spacer[20],Graphics[{Inset[#1,#2]&@@@Thread@{{,,},CirclePoints[3]},GraphicsComplex[CirclePoints[3],{Text["$100",2,{0,2}],Text["¥716",1,{0,-2}],Text["€88.3",3,{0,-2}],Text["$95.3",2,{0,3.5}]}],{Arrowheads[0.08],Arrow[{{0.27,0.7},{0.86,0}}],Arrow[{{0.55,-0.12},{-0.55,-0.12}}],Arrow[{{-0.86,0},{-0.3,0.46}}]},Text[Style["!?!",Bold,18],{0,0.1}],Text[Style["03/27/2025",10,LightGray,"ConcreteText"],{0,-0.45},{0,1}]},ImageSize->140]}]

S=(∮_(∂Ω))(Subscript[A, μ])d(x^μ)=(∫_Ω)(Subscript[F, μ ν])d,(x^μ)d,(x^ν).

Operational definition: Mathematically, the gauge curvature (Subscript[F, μ ν]) is defined by the commutator of covariant derivatives

which measures the amount of non-commutativity to transport the wavefunction along two distinct spacetime directions μ and ν.

Graphics[{{AbsoluteThickness[1],Arrowheads[0.06],Arrow[{{0,0},{1,0}}],Arrow[{{0,0},{0,1}}]},{Text["\!\(x\^μ\)",{1,0},{-1.5,0}],Text["\!\(x\^ν\)",{0,1},{-0.5,-0.5}]},Translate[Block[{a=0.5},{{Lighter[Purple,0.85],Rectangle[{0,0},{a,a}]},{Purple,Text["Ω",{a/2,a/2}]},Arrowheads[{{0.08,0.6}}],{Red,Arrow[{{0,0},{a,0}}],Arrow[{{a,0},{a,a}}],Text["\!\(δ,x\^μ\)",{a/2,0},{0,1.2}],Text["\!\(δ,x\^ν\)",{a,a/2},{-1.5,0}]},{Blue,Arrow[{{0,0},{0,a}}],Arrow[{{0,a},{a,a}}],Text["\!\(δ,x\^μ\)",{a/2,a},{0,-1}],Text["\!\(δ,x\^ν\)",{0,a/2},{1.2,0}]},
{Point[{{0,0},{a,a}}],Text["\!\(ψ(x)\)",{0,0},{1,1}],Text["\!\(ψ(x+δ,x\^μ+δ,x\^ν)\)",{a,a},{-0.9,-1}]}}],{0.3,0.3}]},ImageSize->170]

Using Eq. (), prove Eq. () by comparing the above two paths to transport ψ(x) to ψ(x+δ,(x^μ)+δ,(x^ν)).

Red path:

ψ(x+δ,(x^μ)+δ,(x^ν))=(e^((i/ℏ)(Subscript[A, ν])(x+δ,(x^μ))δ,(x^ν)))(e^(δ,(x^ν)(Subscript[D, ν])))ψ(x+δ,(x^μ))
=(e^((i/ℏ)(Subscript[A, ν])(x+δ,(x^μ))δ,(x^ν)))(e^((i/ℏ)(Subscript[A, μ])(x)δ,(x^μ)))(e^(δ,(x^μ)(Subscript[D, μ])))(e^(δ,(x^ν)(Subscript[D, ν])))ψ(x)
=exp((i/ℏ)((Subscript[A, ν])(x)δ,(x^ν)+(Subscript[A, μ])(x)δ,(x^μ)+(∂_μ)(Subscript[A, ν])(x)δ,(x^μ)δ,(x^ν)+…))exp(δ,(x^μ)(Subscript[D, μ])+δ,(x^ν)(Subscript[D, ν])+(1/2)δ,(x^μ)δ,(x^ν)[(Subscript[D, μ]),(Subscript[D, ν])]+…)ψ(x)

Blue path:

ψ(x+δ,(x^μ)+δ,(x^ν))=(e^((i/ℏ)(Subscript[A, μ])(x+δ,(x^ν))δ,(x^μ)))(e^(δ,(x^μ)(Subscript[D, μ])))ψ(x+δ,(x^ν))
=(e^((i/ℏ)(Subscript[A, μ])(x+δ,(x^ν))δ,(x^μ)))(e^((i/ℏ)(Subscript[A, ν])(x)δ,(x^ν)))(e^(δ,(x^ν)(Subscript[D, ν])))(e^(δ,(x^μ)(Subscript[D, μ])))ψ(x)
=exp((i/ℏ)((Subscript[A, μ])(x)δ,(x^μ)+(Subscript[A, ν])(x)δ,(x^ν)+(∂_ν)(Subscript[A, μ])(x)δ,(x^μ)δ,(x^ν)+…))exp(δ,(x^ν)(Subscript[D, ν])+δ,(x^μ)(Subscript[D, μ])+(1/2)δ,(x^ν)δ,(x^μ)[(Subscript[D, ν]),(Subscript[D, μ])]+…)ψ(x)

exp((i/ℏ)((Subscript[A, ν])(x)δ,(x^ν)+(Subscript[A, μ])(x)δ,(x^μ)+(∂_μ)(Subscript[A, ν])(x)δ,(x^μ)δ,(x^ν)+…))exp(δ,(x^μ)(Subscript[D, μ])+δ,(x^ν)(Subscript[D, ν])+(1/2)δ,(x^μ)δ,(x^ν)[(Subscript[D, μ]),(Subscript[D, ν])]+…)=exp((i/ℏ)((Subscript[A, μ])(x)δ,(x^μ)+(Subscript[A, ν])(x)δ,(x^ν)+(∂_ν)(Subscript[A, μ])(x)δ,(x^μ)δ,(x^ν)+…))exp(δ,(x^ν)(Subscript[D, ν])+δ,(x^μ)(Subscript[D, μ])+(1/2)δ,(x^ν)δ,(x^μ)[(Subscript[D, ν]),(Subscript[D, μ])]+…).

exp((i/ℏ)((∂_μ)(Subscript[A, ν])-(∂_ν)(Subscript[A, μ]))δ,(x^μ)δ,(x^ν)+…)=exp(-δ,(x^μ)δ,(x^ν)[(Subscript[D, μ]),(Subscript[D, ν])]+…).

δ,S=((∂_μ)(Subscript[A, ν])-(∂_ν)(Subscript[A, μ]))δ,(x^μ)δ,(x^ν),

-(i/ℏ)(Subscript[F, μ ν])=[(Subscript[D, μ]),(Subscript[D, ν])]

Physical meaning: In electromagnetism, (Subscript[F, μ ν]) corresponds to the electromagnetic field strength tensor,

Derive Eq. () from Eq. ().

[(Subscript[D, μ]),(Subscript[D, ν])]ψ=((Subscript[D, μ])(Subscript[D, ν])-(Subscript[D, ν])(Subscript[D, μ]))ψ
=((∂_μ)-(i/ℏ)(Subscript[A, μ]))((∂_ν)-(i/ℏ)(Subscript[A, ν]))ψ-((∂_ν)-(i/ℏ)(Subscript[A, ν]))((∂_μ)-(i/ℏ)(Subscript[A, μ]))ψ
=(∂_μ)((∂_ν)ψ-(i/ℏ)(Subscript[A, ν])ψ)-(i/ℏ)(Subscript[A, μ])((∂_ν)ψ-(i/ℏ)(Subscript[A, ν])ψ)-(∂_ν)((∂_μ)ψ-(i/ℏ)(Subscript[A, μ])ψ)+(i/ℏ)(Subscript[A, ν])((∂_μ)ψ-(i/ℏ)(Subscript[A, μ])ψ)
=(∂_μ)(∂_ν)ψ-(i/ℏ)(∂_μ)(Subscript[A, ν])ψ-(i/ℏ)(Subscript[A, ν])(∂_μ)ψ-(i/ℏ)(Subscript[A, μ])(∂_ν)ψ-(1/(ℏ^2))(Subscript[A, μ])(Subscript[A, ν])ψ-(∂_ν)(∂_μ)ψ+(i/ℏ)(∂_ν)(Subscript[A, μ])ψ+(i/ℏ)(Subscript[A, μ])(∂_ν)ψ+(i/ℏ)(Subscript[A, ν])(∂_μ)ψ+(1/(ℏ^2))(Subscript[A, ν])(Subscript[A, μ])ψ
=-(i/ℏ)((∂_μ)(Subscript[A, ν])-(∂_ν)(Subscript[A, μ]))ψ.

[(Subscript[D, μ]),(Subscript[D, ν])]ψ=-(i/ℏ) (Subscript[F, μ ν])ψ,

(Subscript[F, μ ν])=(∂_μ)(Subscript[A, ν])-(∂_ν)(Subscript[A, μ]).

Electric field: E=((E^1),(E^2),(E^3)), with (E^i):=-(Subscript[F, 0i]).

Magnetic field: B=((B^1),(B^2),(B^3)), with (B^i):=(1/2)(ϵ^(i j k))(Subscript[F, j k]).

Check that E and B are gauge invariant. Therefore, they are physical observables.

Φ->Φ-(∂_t)χ,
A→A+∇χ,

E=-∇Φ-(∂_t)A
→-∇(Φ-(∂_t)χ)-(∂_t)(A+∇χ)
=-∇Φ+∇(∂_t)χ-(∂_t)A-(∂_t)∇χ
=-∇Φ-(∂_t)A
=E

B=∇×A
→∇×(A+∇χ)
=∇×A+∇×∇χ
=∇×A
=B

Charged Particle in Gauge Field

Schrödinger picture: state evolves in time, operator remains fixed.

i ℏ(∂_t)ψ=(OverHat[H]) ψ.

Heisenberg picture: operator evolves in time, state remains fixed.

i ℏ(∂_t)(OverHat[O])=[(OverHat[O]),(OverHat[H])].

Note: Eq. () assumes (OverHat[O]) has no explicit time dependence, if not, its explicit time derivative will also contribute to the rate of change of (OverHat[O]).

(OverHat[H])=H((OverHat[x]),(OverHat[p]),t)=(1/(2m))((((OverHat[p])-A((OverHat[x]),t)))^2)+Φ((OverHat[x]),t),

(OverHat[x]) is the coordinate operator.

(OverHat[p])=-i ℏ∇ is the momentum operator.

They satisfy the canonical commutation relation

Verify Eq. ().

[(Subscript[(OverHat[x]), i]),(Subscript[(OverHat[p]), j])]ψ=[(Subscript[x, i]),-i ℏ(∂_j)]ψ
=-i ℏ (Subscript[x, i])(∂_j)ψ+i ℏ(∂_j)((Subscript[x, i])ψ)
=-i ℏ (Subscript[x, i])(∂_j)ψ+i ℏ (Subscript[δ, i j])ψ+i ℏ (Subscript[x, i])(∂_j)ψ
=i ℏ (Subscript[δ, i j])ψ.

[(Subscript[(OverHat[x]), i]),(Subscript[(OverHat[p]), j])]=i ℏ (Subscript[δ, i j])𝟙.

(OverHat[Φ])=Φ((OverHat[x]),t) and (OverHat[A])=A((OverHat[x]),t) are operator functions of (OverHat[x]), with explicit time t dependence.

1st order (velocity operator)

(∂_t)(OverHat[x])=(1/(i ℏ))[(OverHat[x]),(OverHat[H])]=(((OverHat[p])-(OverHat[A]))/m).

2nd order (acceleration operator)

(∂_t^2)(OverHat[x])=-(1/m)(∂_t)(OverHat[A])+(1/(i ℏ))[(∂_t)(OverHat[x]),(OverHat[H])]
=(1/m)((OverHat[E])+(1/2)((∂_t)(OverHat[x])×(OverHat[B])-(OverHat[B])×(∂_t)(OverHat[x]))),

where (OverHat[E]) and (OverHat[B]) operators are defined by

(OverHat[E])=-∇(OverHat[Φ])-(∂_t)(OverHat[A]),
(OverHat[B])=∇×(OverHat[A]).

Derive Eq. () and Eq. ().

[(OverHat[x]),((((OverHat[p])-(OverHat[A])))^2)]:

[(OverHat[x]),((((OverHat[p])-(OverHat[A])))^2)]=[(Subscript[(OverHat[x]), i]),((OverHat[(Subscript[p, j])])-(OverHat[(Subscript[A, j])]))((OverHat[(Subscript[p, j])])-(OverHat[(Subscript[A, j])]))]
=((OverHat[(Subscript[p, j])])-(OverHat[(Subscript[A, j])]))[(Subscript[(OverHat[x]), i]),((OverHat[(Subscript[p, j])])-(OverHat[(Subscript[A, j])]))]+[(Subscript[(OverHat[x]), i]),((OverHat[(Subscript[p, j])])-(OverHat[(Subscript[A, j])]))]((OverHat[(Subscript[p, j])])-(OverHat[(Subscript[A, j])]))
=((OverHat[(Subscript[p, j])])-(OverHat[(Subscript[A, j])]))[(Subscript[(OverHat[x]), i]),(OverHat[(Subscript[p, j])])]+[(Subscript[(OverHat[x]), i]),(OverHat[(Subscript[p, j])])]((OverHat[(Subscript[p, j])])-(OverHat[(Subscript[A, j])]))
=((OverHat[(Subscript[p, j])])-(OverHat[(Subscript[A, j])]))i ℏ (Subscript[δ, i j])+i ℏ (Subscript[δ, i j])((OverHat[(Subscript[p, j])])-(OverHat[(Subscript[A, j])]))
=2i ℏ ((OverHat[(Subscript[p, i])])-(Subscript[(OverHat[A]), i]))
=2i ℏ ((OverHat[p])-(OverHat[A])).

[(OverHat[p])-(OverHat[A]),(OverHat[p])-(OverHat[A])]:

[(OverHat[p])-(OverHat[A]),(OverHat[p])-(OverHat[A])]=[(Subscript[(OverHat[p]), i])-(Subscript[(OverHat[A]), i]),(Subscript[(OverHat[p]), j])-(Subscript[(OverHat[A]), j])]
=-[(Subscript[(OverHat[p]), i]),(Subscript[(OverHat[A]), j])]-[(Subscript[(OverHat[A]), i]),(Subscript[(OverHat[p]), j])]
=-[(Subscript[(OverHat[p]), i]),(Subscript[(OverHat[x]), k])](∂_k)(Subscript[(OverHat[A]), j])-(∂_k)(Subscript[(OverHat[A]), i])[(Subscript[(OverHat[x]), k]),(Subscript[(OverHat[p]), j])]
=i ℏ ((Subscript[δ, i k])(∂_k)(Subscript[(OverHat[A]), j])-(∂_k)(Subscript[(OverHat[A]), i])(Subscript[δ, k j]))
=i ℏ ((∂_i)(Subscript[(OverHat[A]), j])-(∂_j)(Subscript[(OverHat[A]), i]))
=i ℏ (ϵ^(i j k))(ϵ^(k l m))(∂_l)(Subscript[(OverHat[A]), m])
=i ℏ (ϵ^(i j k))(Subscript[(OverHat[B]), k]),

where we have introduced (Subscript[(OverHat[B]), i])=(ϵ^(i j k))(∂_j)(Subscript[(OverHat[A]), k]), or (OverHat[B])=∇×(OverHat[A]).

[(OverHat[p])-(OverHat[A]),((((OverHat[p])-(OverHat[A])))^2)]:

[(OverHat[p])-(OverHat[A]),((((OverHat[p])-(OverHat[A])))^2)]=[(Subscript[(OverHat[p]), i])-(Subscript[(OverHat[A]), i]),((Subscript[(OverHat[p]), j])-(Subscript[(OverHat[A]), j]))((Subscript[(OverHat[p]), j])-(Subscript[(OverHat[A]), j]))]
=((Subscript[(OverHat[p]), j])-(Subscript[(OverHat[A]), j]))[(Subscript[(OverHat[p]), i])-(Subscript[(OverHat[A]), i]),(Subscript[(OverHat[p]), j])-(Subscript[(OverHat[A]), j])]+[(Subscript[(OverHat[p]), i])-(Subscript[(OverHat[A]), i]),(Subscript[(OverHat[p]), j])-(Subscript[(OverHat[A]), j])]((Subscript[(OverHat[p]), j])-(Subscript[(OverHat[A]), j]))
=i ℏ (ϵ^(i j k)) (((Subscript[(OverHat[p]), j])-(Subscript[(OverHat[A]), j]))(Subscript[(OverHat[B]), k])+(Subscript[(OverHat[B]), k]) ((Subscript[(OverHat[p]), j])-(Subscript[(OverHat[A]), j])))
=i ℏ (((OverHat[p])-(OverHat[A]))×(OverHat[B])-(OverHat[B])×((OverHat[p])-(OverHat[A]))).

[(OverHat[p])-(OverHat[A]),(OverHat[Φ])]:

[(OverHat[p])-(OverHat[A]),(OverHat[Φ])]=[(OverHat[p]),(OverHat[Φ])]
=[(Subscript[(OverHat[p]), i]),(Subscript[(OverHat[x]), j])](∂_j)(OverHat[Φ])
=-i ℏ (Subscript[δ, i j])(∂_j)(OverHat[Φ])
=-i ℏ ∇(OverHat[Φ]).

1st order

(∂_t)(OverHat[x])=(1/(i ℏ))[(OverHat[x]),(OverHat[H])]
=(1/(i ℏ))[(OverHat[x]),(1/(2m))((((OverHat[p])-(OverHat[A])))^2)+(OverHat[Φ])]
=(1/(2i ℏ m))[(OverHat[x]),((((OverHat[p])-(OverHat[A])))^2)]
=(1/(2i ℏ m))2i ℏ ((OverHat[p])-(OverHat[A]))
=(((OverHat[p])-(OverHat[A]))/m).

2nd order

(∂_t^2)(OverHat[x])=-(1/m)(∂_t)(OverHat[A])+(1/(i ℏ))[(∂_t)(OverHat[x]),(OverHat[H])]
=-(1/m)(∂_t)(OverHat[A])+(1/(i ℏ m))[(OverHat[p])-(OverHat[A]),(1/(2m))((((OverHat[p])-(OverHat[A])))^2)+(OverHat[Φ])]
=(1/m)(-(∂_t)(OverHat[A])+(1/(i ℏ))((1/(2m))[(OverHat[p])-(OverHat[A]),((((OverHat[p])-(OverHat[A])))^2)]+[(OverHat[p])-(OverHat[A]),(OverHat[Φ])]))
=(1/m)(-(∂_t)(OverHat[A])+(1/(i ℏ))(((i ℏ)/(2m))(((OverHat[p])-(OverHat[A]))×(OverHat[B])-(OverHat[B])×((OverHat[p])-(OverHat[A])))-i ℏ ∇(OverHat[Φ])))
=(1/m)(-(∂_t)(OverHat[A])+(1/2)(((((OverHat[p])-(OverHat[A]))/m))×(OverHat[B])-(OverHat[B])×((((OverHat[p])-(OverHat[A]))/m)))-∇(OverHat[Φ]))
=(1/m)((OverHat[E])+(1/2)((∂_t)(OverHat[x])×(OverHat[B])-(OverHat[B])×(∂_t)(OverHat[x]))).

where we introduce (OverHat[E])=-∇(OverHat[Φ])-(∂_t)(OverHat[A]).

m (Overscript[x, ¨])=F=E+(OverDot[x])×B,

Aharonov-Bohm Effect

Setup: Aharonov-Bohm Experiment

Physical Arrangement: Consider a long, thin solenoid carrying a magnetic flux (Subscript[ϕ, B]). Outside the solenoid, the magnetic field B is zero, but the vector potential A is nonzero. For any surface 𝒮 that fully covers the solenoid, we have

(Subscript[ϕ, B])=(∫_𝒮)B·dσ=(∮_(∂𝒮))A·dl.

Block[{sol,gauge,traj},
sol=Block[{lt=0.9},{{EdgeForm[Black],Lighter[Green,lt],Disk[]},{Green,Point[Cases[Tuples[Range[-1,1,0.4],2],p_/;Norm[p]<1]]},{{{Lighter[Green,lt],Table[Translate[#,r],{r,0.08CirclePoints[24]}]},{Green,#}}&@Text["\!\( B\)",{0,0}]}}];
gauge:=Block[{a=0.2,b=0.5,rs},
rs=Table[Piecewise[{{Exp[1/2(2x-1)],x>1/2},{Sqrt[2x],True}}],{x,a,2,0.5}];{Red,Arrowheads[{0.01,#,Graphics[{Line[Offset/@{{-5,3},{0,0},{-5,-3}}]}]}&/@{0,0.5}],Table[Arrow[ReIm[r Exp[I (b Log[r]+Range[0,2π,π/32])]]],{r,rs}],Text["\!\( A\)",{1.5,0.4}]}];
traj={Blue,{Arrowheads[{{0.06,0.53}}],AbsoluteThickness[2],Scale[Arrow[BSplineCurve[Block[{h1=0.15,h2=2.6},{{-5,0},{-4.9,h1},{-2,h2},{2,h2},{4.9,h1},{5,0}}]]],{1,#},{0,0}]&/@{1,-1}},{Text["\!\(𝒞\_1\)",{0,-2.4},{0,1}],Text["\!\(𝒞\_2\)",{0,2.4},{0,-0.8}]},Translate[Block[{a=0.2},{Disk[{0,0},a],AbsoluteThickness[1],Table[Line[ReIm[{1.6,2.2}a Exp[I θ]]],{θ,π/6,2π,π/6}],Text["source",{0,-0.8},{0,1}]}],{-5,0}],Translate[Block[{h=0.8,α=0.3},{Line[{0,#}&/@{-h,h}],AbsoluteThickness[1],Table[Line[{{0,y},{α h/2,y-α h/2}}],{y,-h+α h,h,α h}],Text["screen",{0,-0.8},{0,1}]}],{5,0}]};
Graphics[{sol,gauge,traj},ImageSize->220]]

Interferometry: A beam of electrons is split into two paths that encircle the solenoid in opposite directions and then recombine to produce an interference pattern.

Key idea: Even when B=0 outside the solenoid, the vector potential A influences the phase of the wavefunction.

When an electron travels along a path 𝒞, the wavefunction acquires a Berry phase:

ψ(Overscript[→, 𝒞])ψ (e^(i (Subscript[Θ, 𝒞])))=ψ exp(((i q)/ℏ)(∫_𝒞)A·dx),

where q=-e is recovered to represent the electron charge.

The phase difference between the two paths is

ΔΘ=(Subscript[Θ, (Subscript[𝒞, 1])])-(Subscript[Θ, (Subscript[𝒞, 2])])=(q/ℏ)((∫_(Subscript[𝒞, 1]))A·dx-(∫_(Subscript[𝒞, 2]))A·dx).

By applying Stokes’ theorem over the surface 𝒮 enclosed by the loop 𝒞=(Subscript[𝒞, 1])-(Subscript[𝒞, 2])=∂𝒮,

ΔΘ=(q/ℏ)(∫_𝒮)(∇×A)·dσ=((q (Subscript[ϕ, B]))/ℏ),

where (Subscript[ϕ, B]) is the magnetic flux through 𝒮 (which equals to the flux inside the solenoid as long as 𝒮 covers the solenoid fully).

This phase shift manifests as a shift in the interference fringes when the two parts of the beam are recombined.

Application: Superconducting Quantum Interference Device (SQUID)

A SQUID consists of a superconducting loop containing a Josephson junction (serving as the screen) and exploit quantum interference to detect extremely subtle changes in magnetic flux inside the loop.

By harnessing the quantum-level sensitivity of the Aharonov-Bohm (AB) effect, SQUIDs can measure magnetic fields as faint as 5×(10^(-18))T at microscopic scales.

SQUIDs also play a pivotal role in quantum computing, as an approach towards superconducting qubits.

Question: What Is Physical about Gauge Fields?

Gauge Potentials vs. Field Strengths: Traditionally, one might think only the fields E and B are physical since they are gauge invariant and can be measured directly by forces. However, the AB effect shows that the potentials Φ and A also have direct physical consequences—they affect the phase of a quantum wavefunction.

Holonomy and Berry Phase: The Berry phase around any closed loop is gauge invariant, and should be physical. All such closed-loop Berry phases (aka. the holonomies) form the complete set of physical observables of a gauge theory. The AB phase is an example of a holonomy: the phase accumulated around a closed loop depends on the curvature (here, the magnetic flux (Subscript[ϕ, B])) enclosed by the loop.

Uniform Magnetic Field

Classical Dynamics

Cyclotron Motion

m - particle mass,

e - particle charge.

E=0, B=B (e^z).

Block[{a=0.15,R=2.6,r=1.4,θ=-π/6},
Graphics[{{Green,AbsolutePointSize[3],Translate[{Circle[{0,0},a],Point[{0,0}]},#]&/@Cases[Tuples[Range[-#,#]&@Ceiling[R],2].{{1,0},{-1/2,Sqrt[3]/2}},p_/;Norm[p]<R],Text["\!\( B\)",{1.9,0.9}]},{Black,Arrowheads[{0.01,#/4,Graphics[Line[Offset/@{{-4,3},{0,0},{-4,-3}}]]}&/@Range[0,3]],Arrow[r ReIm@Exp[I Range[2π,0,-π/32]]],Dashed,Line[ReIm@{0,r Exp[I θ]}],Text["\!\(r\_c\)",ReIm[r/2 Exp[I θ]],{0.8,0.3}],Text["\!\(v\_c\)",{0,r},{0,1}],Point[{0,0}],Text["\!\(ξ\)",{0,0},{2.5,0}]}},ImageSize->120]]

(OverDot[x])=(Subscript[v, c])(cos((Subscript[ω, c])t)(e^x)-sin((Subscript[ω, c])t)(e^y)),
x=(Subscript[r, c])(sin((Subscript[ω, c])t)(e^x)+cos((Subscript[ω, c])t)(e^y))+ξ,

Demonstrate Eq. () by solving the equation of motion Eq. ().

(Overscript[x, ¨])=((q B)/m) (OverDot[x])×(e^z).

(d/(dt))(OverDot[x])=((q B)/m)(OverDot[y]),
(d/(dt))(OverDot[y])=-((q B)/m)(OverDot[x]).

((d^2)/(d(t^2)))(OverDot[x])=-(((((q B)/m)))^2)(OverDot[x]),

(OverDot[x])=(Subscript[v, c])cos((Subscript[ω, c])t),

(OverDot[y])=(m/(q B))(d/(dt))(OverDot[x])=(1/(Subscript[ω, c]))(d/(dt))(OverDot[x])=-(Subscript[v, c])sin((Subscript[ω, c])t).

(OverDot[x])=(Subscript[v, c])(cos((Subscript[ω, c])t)(e^x)-sin((Subscript[ω, c])t)(e^y)).

x=∫dt (OverDot[x])=(Subscript[r, c])(sin((Subscript[ω, c])t)(e^x)+cos((Subscript[ω, c])t)(e^y))+ξ,

(Subscript[ω, c]) - cyclotron frequency:

(Subscript[v, c]) - cyclotron velocity, set by the initial velocity of the particle,

(Subscript[r, c]) - cyclotron radius,

ξ=(Subscript[ξ, x])(e^x)+(Subscript[ξ, y])(e^y) - guiding center (the center of the cyclotron motion). It can be reconstructed from

where π=m (OverDot[x]) denotes the kinetic momentum.

Verify Eq. ().

x-((Subscript[r, c])/(Subscript[v, c]))(e^z)×(OverDot[x])
=(Subscript[r, c])(sin((Subscript[ω, c])t)(e^x)+cos((Subscript[ω, c])t)(e^y))+ξ-((Subscript[r, c])/(Subscript[v, c]))(e^z)×(Subscript[v, c])(cos((Subscript[ω, c])t)(e^x)-sin((Subscript[ω, c])t)(e^y))
=(Subscript[r, c])(sin((Subscript[ω, c])t)(e^x)+cos((Subscript[ω, c])t)(e^y)-cos((Subscript[ω, c])t)(e^z)×(e^x)+sin((Subscript[ω, c])t)(e^z)×(e^y))+ξ
=(Subscript[r, c])(sin((Subscript[ω, c])t)(e^x)+cos((Subscript[ω, c])t)(e^y)-cos((Subscript[ω, c])t)(e^y)-sin((Subscript[ω, c])t)(e^x))+ξ
=ξ.

ξ=x-((Subscript[r, c])/(Subscript[v, c]))(e^z)×(OverDot[x])
=x-((Subscript[r, c])/(m (Subscript[v, c])))(e^z)×m(OverDot[x])
=x-(1/(q B))(e^z)×π.

Hall Effect

E=E (e^y), B=B (e^z).

Block[{a=0.15,R=2.6,h=0.5},
Graphics[{{Green,AbsolutePointSize[3],Translate[{Circle[{0,0},a],Point[{0,0}]},#]&/@Cases[Tuples[Range[-#,#]&@Ceiling[R],2].{{1,0},{-1/2,Sqrt[3]/2}},p_/;Norm[p]<R],Text["\!\( B\)",{1.55,1.22}]},
{Blue,Arrowheads[0.07],Translate[Arrow[{0,Sqrt[3]#/2}&/@{-R,R}],{#,0}]&/@Range[-#,#]&@Floor[R],Text["\!\( E\)",{-1.68,1.9}]},{Black,Arrowheads[{0.01,#/3,Graphics[Line[Offset/@{{-4,3},{0,0},{-4,-3}}]]}&/@Range[1,2]],Translate[{Arrow[{0.82#,0}&/@{-R,R}],Text["\!\(v\_d\)",{0.24R,0},{0,1}]},{0,h}]}},ImageSize->120]]

(OverDot[x])=((E×B)/(B^2))=(Subscript[v, d])(e^x).

Drift velocity of charge:

Corresponding current density:

j=n e (Subscript[v, d])(e^x),

where

n - carrier density, number of charge carriers per unit area.

e - carrier charge.

Justify Eq. ().

dx=(Subscript[v, d])dt.

Block[{Lx=2,Ly=1,dt=0.5},
Graphics[{{GrayLevel[0.9],Rectangle[{-Lx,-Ly},{Lx,Ly}],Lighter[Red,0.8],Rectangle[{-Lx,-Ly},{dt,Ly}],Lighter[Red,0.6],Rectangle[{-Lx,-Ly},{0,Ly}]},{Arrowheads[0.05],Red,Translate[Arrow[{#,0}&/@{0,dt}],{0,# Ly/4}]&/@Range[-3,3]},Translate[{{AbsoluteThickness[1],Arrowheads[{-0.001(-1)^#,#,Graphics[Line[Offset/@{{-4,-3},{0,0},{-4,3}}]]}&/@{0,1}],Arrow[{0,#}&/@{-Ly,Ly}]},{{GrayLevel[0.9],Table[Translate[#,r],{r,0.07CirclePoints[32]}]},#}&@Text["\!\(L\_y\)"]},{0.8Lx,0}],{Black,Translate[Line[{#,0}&/@{-Lx,Lx}],{0,#}]&/@{-Ly,Ly},Dashed,Red,Translate[Line[{0,#}&/@{-Ly,Ly}],{#,0}]&/@{0,dt}},Translate[{Text["\!\(v\_d \[DifferentialD]t\)",{dt/2,0},{0,1}]},{0,-Ly}]},ImageSize->160]]

dq=e n (Subscript[L, y]) dx,

(Subscript[I, x])=((dq)/(dt))=e n (Subscript[L, y])((dx)/(dt))=n e (Subscript[v, d]) (Subscript[L, y]).

(Subscript[j, x])=((Subscript[I, x])/(Subscript[L, y]))=n e (Subscript[v, d]).

Hall conductivity:

Hall resistivity:

(Subscript[ρ, H]):=((Subscript[E, y])/(Subscript[j, x]))=(B/(n e)).

Block[{β=80.,μ=1.,σH},
σH[B_?NumericQ]:=Block[{c=5.,nmin,nmax},
{nmin,nmax}=(2{-1,1}c-B β+2β μ)/(2B β);
nmin=Max[0,Floor[nmin]];
nmax=Ceiling[nmax];
nmin+If[nmax-nmin<10,NSum[LogisticSigmoid[β(μ-(n+1/2)B)],{n,nmin,nmax}],1/2+NIntegrate[LogisticSigmoid[β(μ-(n+1/2)B)],{n,nmin,nmax}]]];
Plot[{B,1/σH[B]},{B,0.001,1},GridLines->{None,1/Range[6]},PlotStyle->{Directive[AbsoluteThickness[1.2],Dashed,LightGray],Black},FrameTicks->{{Automatic,{1/#,If[#==1,"1",If[MemberQ[{2,3,6},#],Style[StringTemplate["1/`1`"][#],10],""]]}&/@Range[6]},{Automatic,Automatic}},FrameLabel->{"\!\( B\) [a.u.]","\!\(ρ\_H\) [\!\(h/e\^2\)]"},ImageSize->180]]

Classical expectation: given fixed carrier (electron) density n, (Subscript[ρ, H])∝B should scale linearly with B.

Quantum Hall effect: when B is large enough, (Subscript[σ, H])=(Subscript[ρ, H]^(-1)) exhibits steps at quantized values:

h - Planck constant,

e - electron charge.

Landau Level Quantization

Gauge Field

B=B (e^z)=∇×A.

The vector potential (gauge field) A circulates in the x-y plane,

Block[{a=0.15,R=2.6,θ=-π/4},
Graphics[{{Green,AbsolutePointSize[3],Translate[{Circle[{0,0},a],Point[{0,0}]},#]&/@Cases[Tuples[Range[-#,#]&@Ceiling[R],2].{{1,0},{-1/2,Sqrt[3]/2}},p_/;Norm[p]<R],Text["\!\( B\)",{1.9,0.9}]},{Red,Arrowheads[{0.01,#/4,Graphics[Line[Offset/@{{-4,3},{0,0},{-4,-3}}]]}&/@Range[0,3]],Table[Arrow[r ReIm@Exp[I (Range[0,2π,π/32]+θ)]],{r,Sqrt[Range[1,3]]}],Text["\!\( A\)",ReIm[-I Sqrt[3]Exp[I θ]],{1,1}]}},ImageSize->120]]

known as the symmetric gauge.

Verify Eq. () reproduces Eq. ().

∇×A=((∂_x)(Subscript[A, y])-(∂_y)(Subscript[A, x]))(e^z)
=(B/2)((∂_x)x+(∂_y)y)(e^z)
=(B/2)(1+1)(e^z)
=B (e^z)=B.

However, the gauge choice is not unique. For example, the following gauge choice is also valid:

A=((Subscript[A, x]),(Subscript[A, y]))=(0, B x),

known as the Landau gauge.

David Tong. The Quantum Hall Effect (TIFR Infosys Lectures), (2016).

Hamiltonian

(OverHat[H])=-((ℏ^2)/(2m))(D^2)=-((ℏ^2)/(2m))(((∇-((i e)/ℏ)(OverHat[A])))^2).

Unit choice:

charge e=1,

mass m=1,

Planck constant ℏ=1.

Eq. () can be simplified to

(OverHat[p])=-i ∇=(-i (∂_x),-i(∂_y)) - canonical momentum operator.

Canonical commutation relation with coordinate operator (OverHat[x])=((OverHat[x]),(OverHat[y])):

[(Subscript[(OverHat[x]), i]),(Subscript[(OverHat[p]), j])]=i     (fori,j=x,y).

(OverHat[A])=A((OverHat[x]))=((Subscript[(OverHat[A]), x]),(Subscript[(OverHat[A]), y])) - potential momentum operator (electromagnetic vector potential). Under symmetric gauge Eq. (),

(Subscript[(OverHat[A]), x])=-(B/2)(OverHat[y]), (Subscript[(OverHat[A]), y])=(B/2)(OverHat[x]).

(OverHat[π])=((Subscript[(OverHat[π]), x]),(Subscript[(OverHat[π]), y]))  - kinetic momentum operator,

or, in terms of components

(Subscript[(OverHat[π]), x])=(Subscript[(OverHat[p]), x])-(Subscript[(OverHat[A]), x])=-i (∂_x)-(Subscript[(OverHat[A]), x]),
(Subscript[(OverHat[π]), y])=(Subscript[(OverHat[p]), y])-(Subscript[(OverHat[A]), y])=-i (∂_y)-(Subscript[(OverHat[A]), y]).

They satisfy the following commutation relation

which follows from the definition of gauge curvature in Eq. ().

Verify Eq. ().

[(Subscript[(OverHat[π]), x]),(Subscript[(OverHat[π]), y])]=[-i (∂_x)-(Subscript[A, x]),-i (∂_y)-(Subscript[A, y])]
=-[(∂_x),(∂_y)]+i [(∂_x),(Subscript[A, y])]+i [(Subscript[A, x]),(∂_y)]+[(Subscript[A, x]),(Subscript[A, y])]
=0+i (∂_x)(Subscript[A, y])-i (∂_y)(Subscript[A, x])+0
=i ((∂_x)(Subscript[A, y])-(∂_y)(Subscript[A, x]))
=i B.

They also inherit the commutation relation with the coordinate operator,

[(Subscript[(OverHat[x]), i]),(Subscript[(OverHat[π]), j])]=i     (fori,j=x,y).

Guiding Center

Under symmetric gauge Eq. (),

(OverHat[ξ])=(1/B)((OverHat[p])+(OverHat[A]))×(e^z),

Verify Eq. ().

(OverHat[ξ])=(OverHat[x])-(1/B)(e^z)×(OverHat[π])
=(OverHat[x])(e^x)+(OverHat[y])(e^y)-(1/B)(e^z)×(((Subscript[(OverHat[p]), x])-(Subscript[(OverHat[A]), x]))(e^x)+((Subscript[(OverHat[p]), y])-(Subscript[(OverHat[A]), y]))(e^y))
=(OverHat[x])(e^x)+(OverHat[y])(e^y)-(1/B)(((Subscript[(OverHat[p]), x])+(B/2)(OverHat[y]))(e^z)×(e^x)+((Subscript[(OverHat[p]), y])-(B/2)(OverHat[x]))(e^z)×(e^y))
=(1/B)(B(OverHat[x])(e^x)+B(OverHat[y])(e^y)-((Subscript[(OverHat[p]), x])+(B/2)(OverHat[y]))(e^y)+((Subscript[(OverHat[p]), y])-(B/2)(OverHat[x]))(e^x))
=(1/B)(((Subscript[(OverHat[p]), y])+(B/2)(OverHat[x]))(e^x)-((Subscript[(OverHat[p]), x])-(B/2)(OverHat[y]))(e^y)).

(1/B)((OverHat[p])+(OverHat[A]))×(e^z)
=(1/B)(((Subscript[(OverHat[p]), x])+(Subscript[(OverHat[A]), x]))(e^x)+((Subscript[(OverHat[p]), y])+(Subscript[(OverHat[A]), y]))(e^y))×(e^z)
=(1/B)(((Subscript[(OverHat[p]), x])-(B/2)(OverHat[y]))(e^x)×(e^z)+((Subscript[(OverHat[p]), y])+(B/2)(OverHat[x]))(e^y)×(e^z))
=(1/B)(-((Subscript[(OverHat[p]), x])-(B/2)(OverHat[y]))(e^y)+((Subscript[(OverHat[p]), y])+(B/2)(OverHat[x]))(e^x)).

(OverHat[ξ])=(1/B)((OverHat[p])+(OverHat[A]))×(e^z).

or, in terms of components

(Subscript[(OverHat[ξ]), x])=(OverHat[x])+(1/B)(Subscript[(OverHat[π]), y])=(1/B)((Subscript[(OverHat[p]), y])+(Subscript[(OverHat[A]), y])),
(Subscript[(OverHat[ξ]), y])=(OverHat[y])-(1/B)(Subscript[(OverHat[π]), x])=-(1/B)((Subscript[(OverHat[p]), x])+(Subscript[(OverHat[A]), x])).

They satisfy the following commutation relation

Prove Eq. ().

[(Subscript[(OverHat[ξ]), x]),(Subscript[(OverHat[ξ]), y])]=[(OverHat[x])+(1/B)(Subscript[(OverHat[π]), y]),(OverHat[y])-(1/B)(Subscript[(OverHat[π]), x])]
=[(OverHat[x]),(OverHat[y])]+(1/B)[(Subscript[(OverHat[π]), y]),(OverHat[y])]-(1/B)[(OverHat[x]),(Subscript[(OverHat[π]), x])]-(1/(B^2))[(Subscript[(OverHat[π]), y]),(Subscript[(OverHat[π]), x])]
=0-(1/B)[(OverHat[y]),(Subscript[(OverHat[π]), y])]-(1/B)[(OverHat[x]),(Subscript[(OverHat[π]), x])]+(1/(B^2))[(Subscript[(OverHat[π]), x]),(Subscript[(OverHat[π]), y])]
=-(i/B)-(i/B)+(1/(B^2))(i B),
=(1/(i B)).

Guiding center and kinetic momentum operators commute.

Prove Eq. ().

[(Subscript[(OverHat[ξ]), x]),(Subscript[(OverHat[π]), x])]=[(OverHat[x])+(1/B)(Subscript[(OverHat[π]), y]),(Subscript[(OverHat[π]), x])]
=[(OverHat[x]),(Subscript[(OverHat[π]), x])]+(1/B)[(Subscript[(OverHat[π]), y]),(Subscript[(OverHat[π]), x])]
=i+(1/B)(-i B)=0.

[(Subscript[(OverHat[ξ]), y]),(Subscript[(OverHat[π]), y])]=[(OverHat[y])-(1/B)(Subscript[(OverHat[π]), x]),(Subscript[(OverHat[π]), y])]
=[(OverHat[y]),(Subscript[(OverHat[π]), y])]-(1/B)[(Subscript[(OverHat[π]), x]),(Subscript[(OverHat[π]), y])]
=i-(1/B)(i B)=0.

[(Subscript[(OverHat[ξ]), x]),(Subscript[(OverHat[π]), y])]=[(OverHat[x])+(1/B)(Subscript[(OverHat[π]), y]),(Subscript[(OverHat[π]), y])]
=[(OverHat[x]),(Subscript[(OverHat[π]), y])]+(1/B)[(Subscript[(OverHat[π]), y]),(Subscript[(OverHat[π]), y])]
=0+0=0.

[(Subscript[(OverHat[ξ]), y]),(Subscript[(OverHat[π]), x])]=[(OverHat[y])-(1/B)(Subscript[(OverHat[π]), x]),(Subscript[(OverHat[π]), x])]
=[(OverHat[y]),(Subscript[(OverHat[π]), x])]-(1/B)[(Subscript[(OverHat[π]), x]),(Subscript[(OverHat[π]), x])]
=0+0=0.

[(Subscript[(OverHat[ξ]), x]),(Subscript[(OverHat[π]), x])]=[(Subscript[(OverHat[ξ]), y]),(Subscript[(OverHat[π]), y])]=[(Subscript[(OverHat[ξ]), x]),(Subscript[(OverHat[π]), y])]=[(Subscript[(OverHat[ξ]), y]),(Subscript[(OverHat[π]), x])]=0.

Annihilation and Creation Operators

Cyclotron annihilation and creation operators:

such that

(SuperDagger[(OverHat[a])])(OverHat[a])+(1/2)=(1/(2B))((OverHat[π])^2)=(1/(2B))((((OverHat[p])-(OverHat[A])))^2).

Verify Eq. ().

(SuperDagger[(OverHat[a])])(OverHat[a])+(1/2)=(1/(2B))((Subscript[(OverHat[π]), x])-i (Subscript[(OverHat[π]), y]))((Subscript[(OverHat[π]), x])+i (Subscript[(OverHat[π]), y]))+(1/2)
=(1/(2B))((Subscript[(OverHat[π]), x]^2)+(Subscript[(OverHat[π]), y]^2)-i (Subscript[(OverHat[π]), y])(Subscript[(OverHat[π]), x])+i (Subscript[(OverHat[π]), x])(Subscript[(OverHat[π]), y]))+(1/2)
=(1/(2B))((Subscript[(OverHat[π]), x]^2)+(Subscript[(OverHat[π]), y]^2))+(1/(2B))i [(Subscript[(OverHat[π]), x]),(Subscript[(OverHat[π]), y])]+(1/2)
=(1/(2B))((OverHat[π])^2)+(1/(2B))i (i B)+(1/2)
=(1/(2B))((OverHat[π])^2)-(1/2)+(1/2)
=(1/(2B))((OverHat[π])^2).

(SuperDagger[(OverHat[a])])(OverHat[a])+(1/2)=(1/(2B))((OverHat[π])^2)=(1/(2B))((((OverHat[p])-(OverHat[A])))^2).

Guiding center annihilation and creation operators:

such that

(SuperDagger[(OverHat[b])])(OverHat[b])+(1/2)=(B/2)((OverHat[ξ])^2)=(1/(2B))((((OverHat[p])+(OverHat[A])))^2).

Verify Eq. ().

(SuperDagger[(OverHat[b])])(OverHat[b])+(1/2)=(B/2)((Subscript[(OverHat[ξ]), x])+i (Subscript[(OverHat[ξ]), y]))((Subscript[(OverHat[ξ]), x])-i (Subscript[(OverHat[ξ]), y]))+(1/2)
=(B/2)((Subscript[(OverHat[ξ]), x]^2)+(Subscript[(OverHat[ξ]), y]^2)+i (Subscript[(OverHat[ξ]), y])(Subscript[(OverHat[ξ]), x])-i (Subscript[(OverHat[ξ]), x])(Subscript[(OverHat[ξ]), y]))+(1/2)
=(B/2)((Subscript[(OverHat[ξ]), x]^2)+(Subscript[(OverHat[ξ]), y]^2))-(B/2)i [(Subscript[(OverHat[ξ]), x]),(Subscript[(OverHat[ξ]), y])]+(1/2)
=(B/2)((OverHat[ξ])^2)-(B/2)(i/(i B))+(1/2)
=(B/2)((OverHat[ξ])^2)-(1/2)+(1/2)
=(B/2)((OverHat[ξ])^2).

(SuperDagger[(OverHat[b])])(OverHat[b])+(1/2)=(B/2)((OverHat[ξ])^2)
=(B/2)(1/(B^2))(((((OverHat[p])+(OverHat[A]))×(e^z)))^2)
=(1/(2B))((((OverHat[p])+(OverHat[A])))^2).

Prove Eq. ().

[(OverHat[a]),(SuperDagger[(OverHat[a])])]=(1/(2B))[(Subscript[(OverHat[π]), x])+i (Subscript[(OverHat[π]), y]),(Subscript[(OverHat[π]), x])-i (Subscript[(OverHat[π]), y])]
=(1/(2B))([(Subscript[(OverHat[π]), x]),(Subscript[(OverHat[π]), x])]+i [(Subscript[(OverHat[π]), y]),(Subscript[(OverHat[π]), x])]-i [(Subscript[(OverHat[π]), x]),(Subscript[(OverHat[π]), y])]+[(Subscript[(OverHat[π]), y]),(Subscript[(OverHat[π]), y])])
=(1/(2B))(0-i [(Subscript[(OverHat[π]), x]),(Subscript[(OverHat[π]), y])]-i [(Subscript[(OverHat[π]), x]),(Subscript[(OverHat[π]), y])]+0)
=(1/(2B))(-2i)(i B)
=1.

[(OverHat[b]),(SuperDagger[(OverHat[b])])]=(B/2)[(Subscript[(OverHat[ξ]), x])-i (Subscript[(OverHat[ξ]), y]),(Subscript[(OverHat[ξ]), x])+i (Subscript[(OverHat[ξ]), y])]
=(B/2)([(Subscript[(OverHat[ξ]), x]),(Subscript[(OverHat[ξ]), x])]-i [(Subscript[(OverHat[ξ]), y]),(Subscript[(OverHat[ξ]), x])]+i [(Subscript[(OverHat[ξ]), x]),(Subscript[(OverHat[ξ]), y])]+[(Subscript[(OverHat[ξ]), y]),(Subscript[(OverHat[ξ]), y])])
=(B/2)(0+i [(Subscript[(OverHat[ξ]), x]),(Subscript[(OverHat[ξ]), y])]+i [(Subscript[(OverHat[ξ]), x]),(Subscript[(OverHat[ξ]), y])]+0)
=(B/2)(2i)(1/(i B))
=1.

[(OverHat[a]),(OverHat[b])]=[(OverHat[a]),(SuperDagger[(OverHat[b])])]=[(SuperDagger[(OverHat[a])]),(OverHat[b])]=[(SuperDagger[(OverHat[a])]),(SuperDagger[(OverHat[b])])]=0.

Implication: (OverHat[a]) and (OverHat[b]) are annihilation operators for two independent harmonic oscillator degrees of freedom.

(SuperDagger[(OverHat[a])])(OverHat[a]) and (SuperDagger[(OverHat[b])])(OverHat[b]) are commuting number operators, and can be diagonalized simultaneously. Their eigenvalues correspond to two separate sets of quantum numbers, denoted as n and m respectively:

n,m=0,1,2,…∈ℕ.

Landau Levels

Hamiltonian: using Eq. (), Eq. () can be written as

Angular momentum:

Verify Eq. () under symmetric gauge.

(SuperDagger[(OverHat[b])])(OverHat[b])-(SuperDagger[(OverHat[a])])(OverHat[a])=(1/(2B))((((OverHat[p])+(OverHat[A])))^2)-(1/(2B))((((OverHat[p])-(OverHat[A])))^2)
=(1/(2B))(((OverHat[p])^2)+(OverHat[p])·(OverHat[A])+(OverHat[A])·(OverHat[p])+((OverHat[A])^2))-(1/(2B))(((OverHat[p])^2)-(OverHat[p])·(OverHat[A])-(OverHat[A])·(OverHat[p])+((OverHat[A])^2))
=(1/B)((OverHat[p])·(OverHat[A])+(OverHat[A])·(OverHat[p]))
=(1/B)((Subscript[(OverHat[p]), x])(Subscript[(OverHat[A]), x])+(Subscript[(OverHat[p]), y])(Subscript[(OverHat[A]), y])+(Subscript[(OverHat[A]), x])(Subscript[(OverHat[p]), x])+(Subscript[(OverHat[A]), y])(Subscript[(OverHat[p]), y]))
=(1/B)((Subscript[(OverHat[p]), x])(-(B/2)(OverHat[y]))+(Subscript[(OverHat[p]), y])((B/2)(OverHat[x]))+(-(B/2)(OverHat[y]))(Subscript[(OverHat[p]), x])+((B/2)(OverHat[x]))(Subscript[(OverHat[p]), y]))
=(1/2)(-(Subscript[(OverHat[p]), x])(OverHat[y])+(Subscript[(OverHat[p]), y])(OverHat[x])-(OverHat[y])(Subscript[(OverHat[p]), x])+(OverHat[x])(Subscript[(OverHat[p]), y]))
=(1/2)(-(OverHat[y])(Subscript[(OverHat[p]), x])+(OverHat[x])(Subscript[(OverHat[p]), y])-(OverHat[y])(Subscript[(OverHat[p]), x])+(OverHat[x])(Subscript[(OverHat[p]), y]))
=(OverHat[x])(Subscript[(OverHat[p]), y])-(OverHat[y])(Subscript[(OverHat[p]), x])
=((OverHat[x])×(OverHat[p]))·(e^z)
=(Subscript[(OverHat[L]), z]).

Their common eigenstates are (Ket[n,m]):

which are labeled by two quantum numbers:

n: Landau level index (energy level index),

m: angular momentum index (degeneracy within a Landau level).

The energy levels are quantized

(Subscript[E, n])=B (n+(1/2)),

with n=0,1,2,…∈ℕ.

After restoring the energy unit,

where (Subscript[ω, c])=q B/m is the cyclotron frequency.

Each level is called a Landau level. The n=0 level is called the lowest Landau level (LLL).

The angular momentum is also quantized. After restoring the unit,

Graphics[Table[Point[{m-n,n+1/2}],{n,0,5},{m,0,n+5}],PlotRange->{{-5.5,5.5},{0,6}},GridLines->{Range[-5,5],Range[0,5]+1/2},Frame->True,FrameLabel->{"\!\(L\_\(z,n,m\)/ℏ\)","\!\(E\_n/\(ℏ ω\_c\)\)"},ImageSize->180]

Landau level degeneracy:

In an infinite system, each Landau level is infinitely degenerated.

Argument: The quantum number m=0,1,2,…∈ℕ is unbounded. => Infinitely many orthogonal states (Ket[n,m]) within the same energy level n.

In a realistic system of finite size, the Landau level degeneracy becomes finite, and is determined by the total magnetic flux measured in units of the flux quantum.

Consider electrons confined within a disk of radius R. => The guiding center radius ξ must satisfy ξ≲R.

This puts a constraint on the operator

(SuperDagger[(OverHat[b])])(OverHat[b])+(1/2)=(B/2)((OverHat[ξ])^2)≲((B (R^2))/2),

in terms of its eigenvalues. => m≲B (R^2)/2, restoring the physical units:

m≲((e B (R^2))/(2ℏ)).

A flux quantum is the amount of magnetic flux (Subscript[ϕ, 0]) that induces a 2π Berry phase for an electron braiding around it.

phase=(action/ℏ)=((e (Subscript[ϕ, 0]))/ℏ)=2π,

meaning that

Then Eq. () can be written as

where (Subscript[ϕ, B])=π (R^2) B is the total magnetic flux through the disk.

Thus the Landau level degeneracy is set by (Subscript[ϕ, B])/(Subscript[ϕ, 0]) — the conclusion generalizes to any shape of the system.

Complex Coordinate

the complex coordinate

and the complex derivative

Derive Eq. () from Eq. ().

x=((z+(OverBar[z]))/(Sqrt[2B])), y=((z-(OverBar[z]))/(i(Sqrt[2B]))).

(∂_z)=((∂x)/(∂z))(∂/(∂x))+((∂y)/(∂z))(∂/(∂y))
=(1/(Sqrt[2B]))(∂_x)+(1/(i(Sqrt[2B])))(∂_y)
=(1/(Sqrt[2B]))((∂_x)-i(∂_y)).

(∂_(OverBar[z]))=((∂x)/(∂(OverBar[z])))(∂/(∂x))+((∂y)/(∂(OverBar[z])))(∂/(∂y))
=(1/(Sqrt[2B]))(∂_x)+((-1)/(i(Sqrt[2B])))(∂_y)
=(1/(Sqrt[2B]))((∂_x)+i(∂_y)).

Verify Eq. ().

(OverHat[a])=(1/(Sqrt[2B]))((Subscript[(OverHat[π]), x])+i (Subscript[(OverHat[π]), y]))
=(1/(Sqrt[2B]))(((Subscript[(OverHat[p]), x])-(Subscript[(OverHat[A]), x]))+i ((Subscript[(OverHat[p]), y])-(Subscript[(OverHat[A]), y])))
=(1/(Sqrt[2B]))((-i (∂_x)+(B/2)y)+i (-i(∂_y)-(B/2)x))
=-i (1/2)(Sqrt[(B/2)])(x+i y)-i(1/(Sqrt[2B]))((∂_x)+i (∂_y))
=-i ((1/2)z+(∂_(OverBar[z]))).

(OverHat[b])=(Sqrt[(B/2)])((Subscript[(OverHat[ξ]), x])-i (Subscript[(OverHat[ξ]), y]))
=(Sqrt[(B/2)])(1/B)(((Subscript[(OverHat[p]), y])+(Subscript[(OverHat[A]), y]))+i ((Subscript[(OverHat[p]), x])+(Subscript[(OverHat[A]), x])))
=(1/(Sqrt[2B]))((-i(∂_y)+(B/2)x)+i (-i(∂_x)-(B/2)y))
=(1/2)(Sqrt[(B/2)])(x-i y)+(1/(Sqrt[2B]))((∂_x)-i(∂_y))
=(1/2)(OverBar[z])+(∂_z).

(SuperDagger[(OverHat[a])])=(SuperDagger[(-i ((1/2)z+(∂_(OverBar[z]))))])
=i ((1/2)(OverBar[z])-(∂_z)).

(SuperDagger[(OverHat[b])])=(SuperDagger[((1/2)(OverBar[z])+(∂_z))])
=(1/2)z-(∂_(OverBar[z])).

Wave Functions

((1/2)z+(∂_(OverBar[z])))(Subscript[ψ, 0,0])(z,(OverBar[z]))=((1/2)(OverBar[z])+(∂_z))(Subscript[ψ, 0,0])(z,(OverBar[z]))=0.

Consider the ansatz

(Subscript[ψ, 0,0])(z,(OverBar[z]))=f(z,(OverBar[z]))(e^(-(OverBar[z])z/2)),

Eq. () implies

(∂_z)f(z,(OverBar[z]))=(∂_(OverBar[z]))f(z,(OverBar[z]))=0,

Derive Eq. ().

((1/2)z+(∂_(OverBar[z])))(Subscript[ψ, 0,0])(z,(OverBar[z]))
=((1/2)z+(∂_(OverBar[z])))(f(z,(OverBar[z]))(e^(-(OverBar[z])z/2)))
=(1/2)z f(z,(OverBar[z]))(e^(-(OverBar[z])z/2))+(∂_(OverBar[z]))(f(z,(OverBar[z]))(e^(-(OverBar[z])z/2)))
=(1/2)z f(z,(OverBar[z]))(e^(-(OverBar[z])z/2))+(∂_(OverBar[z]))f(z,(OverBar[z]))(e^(-(OverBar[z])z/2))+f(z,(OverBar[z]))(∂_(OverBar[z]))(e^(-(OverBar[z])z/2))
=(1/2)z f(z,(OverBar[z]))(e^(-(OverBar[z])z/2))+(∂_(OverBar[z]))f(z,(OverBar[z]))(e^(-(OverBar[z])z/2))-(1/2)z f(z,(OverBar[z]))(e^(-(OverBar[z])z/2))
=(∂_(OverBar[z]))f(z,(OverBar[z]))(e^(-(OverBar[z])z/2))=0,

((1/2)(OverBar[z])+(∂_z))(Subscript[ψ, 0,0])(z,(OverBar[z]))
=((1/2)(OverBar[z])+(∂_z))(f(z,(OverBar[z]))(e^(-(OverBar[z])z/2)))
=(1/2)(OverBar[z]) f(z,(OverBar[z]))(e^(-(OverBar[z])z/2))+(∂_z)(f(z,(OverBar[z]))(e^(-(OverBar[z])z/2)))
=(1/2)(OverBar[z]) f(z,(OverBar[z]))(e^(-(OverBar[z])z/2))+(∂_z)f(z,(OverBar[z]))(e^(-(OverBar[z])z/2))+f(z,(OverBar[z]))(∂_z)(e^(-(OverBar[z])z/2))
=(1/2)(OverBar[z]) f(z,(OverBar[z]))(e^(-(OverBar[z])z/2))+(∂_z)f(z,(OverBar[z]))(e^(-(OverBar[z])z/2))-(1/2)(OverBar[z]) f(z,(OverBar[z]))(e^(-(OverBar[z])z/2))
=(∂_z)f(z,(OverBar[z]))(e^(-(OverBar[z])z/2))=0.

(∂_z)f(z,(OverBar[z]))=(∂_(OverBar[z]))f(z,(OverBar[z]))=0.

meaning that f(z,(OverBar[z])) must be independent of both z and (OverBar[z]), i.e. it is a constant function, i.e.

f(z,(OverBar[z]))=const.

Therefore, the vacuum state wave function reads

which is normalized to ensure ∫dzd(OverBar[z])((|(Subscript[ψ, 0,0])(z,(OverBar[z]))|)^2)=1.

Lowest Landau level (LLL): The wave functions with n=0 are

Derive Eq. ().

((1/2)z-(∂_(OverBar[z])))(e^(-(OverBar[z])z/2))
=(1/2)z (e^(-(OverBar[z])z/2))-(∂_(OverBar[z]))(e^(-(OverBar[z])z/2))
=(1/2)z (e^(-(OverBar[z])z/2))-(-(z/2))(e^(-(OverBar[z])z/2))
=z (e^(-(OverBar[z])z/2)),

((((1/2)z-(∂_(OverBar[z]))))^m)(e^(-(OverBar[z])z/2))
=((((1/2)z-(∂_(OverBar[z]))))^(m-1))z (e^(-(OverBar[z])z/2))
=z ((((1/2)z-(∂_(OverBar[z]))))^(m-1))(e^(-(OverBar[z])z/2))
=z ((((1/2)z-(∂_(OverBar[z]))))^(m-2))z (e^(-(OverBar[z])z/2))
=(z^2) ((((1/2)z-(∂_(OverBar[z]))))^(m-2)) (e^(-(OverBar[z])z/2))
=…
=(z^m) (e^(-(OverBar[z])z/2)).

(Subscript[ψ, 0,m])(z,(OverBar[z]))=(1/(Sqrt[m!]))((((SuperDagger[(OverHat[b])])))^m)(Subscript[ψ, 0,0])(z,(OverBar[z]))
=(1/(Sqrt[π m!]))((((1/2)z-(∂_(OverBar[z]))))^m)(e^(-(OverBar[z])z/2))
=(1/(Sqrt[π m!]))(z^m)(e^(-(OverBar[z])z/2)).

The wave functions has circular symmetry, taking the ring shape.

DynamicModule[{cfun},
cfun=Darker[Blend[{RGBColor[0.91,0.36,0.46],RGBColor[1.,0.59,0.4],RGBColor[0.97,0.76,0],RGBColor[0.72,0.8,0.46],RGBColor[0.25,0.73,0.3],RGBColor[0,0.76,0.93],RGBColor[0.43,0.58,0.99],RGBColor[0.72,0.53,0.995],RGBColor[0.91,0.36,0.46]},Mod[#2+1/2,1]],1-#1]&;
Manipulate[
ComplexPlot[Evaluate[Nest[(Conjugate[z] #-D[#,z]/.Conjugate'[z]->0)&,z^m,n] Exp[-Abs[z]^2/2]/Sqrt[π n! m!]],{z,-5-5I,5+5I},Epilog->{White,Dashed,Opacity[0.5],If[m+n>0,Circle[{0,0},Sqrt[(n+m)]],Nothing]},ColorFunction->{cfun[#7,#8]&,None},FrameLabel->{"\!\(Re z=\@\(B/2\) x\)","\!\(Im z=\@\(B/2\) y\)"},PlotLabel->StringTemplate["\!\(\[LeftBracketingBar]`1`,`2`⟩\)"][n,m],ImageSize->180],{{n,0,Style["\!\( n:\)","InlineFormula"]},Range[0,5],ControlType->SetterBar},{{m,6,Style["\!\( m:\)","InlineFormula"]},Range[0,10],ControlType->SetterBar},Paneled->False,SaveDefinitions->True]]

Probability density

((|(Subscript[ψ, 0,m])(z,(OverBar[z]))|)^2)=(1/(π m!))((((OverBar[z])z))^m)(e^(-(OverBar[z])z)),

which distributes around the radius |z|=(Sqrt[m]).

Estimate the radius |z| that maximizes the probability.

p(|z|)=(1/(π m!))((|z|)^(2m))(e^(-((|z|)^2)))

((dp(|z|))/(d|z|))=(2/(π m!))((|z|)^(2m-1))(e^(-((|z|)^2)))(m-((|z|)^2))=0,

D[z^(2m)Exp[-z^2],z]//Simplify

Semi-classical argument: The factor (z^m)=(Sqrt[B/2])(r^m)(e^(i m θ)) implies that the particle accumulates 2π m Berry phase when travelling around a circle of radius r.

To ensure that the wave function is single-valued, m must be an integer, i.e.

2π m∈2π ℤ=>m∈ℤ.

This corresponds to a canonical momentum of

p=((2π m)/(2π r))=(m/r).

States in the LLL should minimize the energy E=(((p-A))^2)/2, which can be achieved if

p=A=((π (r^2) B)/(2π r))=((r B)/2).

Eq. () and Eq. () together sets the optimal radius

(m/r)=p=((r B)/2)=>r=(Sqrt[((2m)/B)]),

or, in terms of z variable, |z| =(Sqrt[B/2])r=(Sqrt[m]) (with m∈ℤ).

Any linear combination of (Ket[0,m]) is still a state in the LLL, which takes the general form of

(∑_m)(Subscript[c, m])(Subscript[ψ, 0,m])(z,(OverBar[z]))=(1/(Sqrt[π m!]))((∑_m)(Subscript[c, m])(z^m))(e^(-(OverBar[z])z/2))=f(z)(e^(-(OverBar[z])z/2)).

Conclusion: LLL wavefunctions are holomorphic functions of  z , multiplied by a Gaussian envelope.

More generally, for higher Landau levels, the wave functions

(Subscript[ψ, n,m])(z,(OverBar[z]))=(1/(Sqrt[π n!m!]))(((((OverBar[z])-(∂_z)))^n)(z^m))(e^(-(OverBar[z])z/2)),

can be written in terms of associated Laguerre polynomials.

Quantum Hall Effect

Filling Landau Levels

(Subscript[E, n])=ℏ (Subscript[ω, c])(n+(1/2))      (n=0,1,2,…).

Level spacing (energy gap): ℏ (Subscript[ω, c]), where (Subscript[ω, c])=e B/m is the cyclotron frequency.

Level degeneracy: Given the total magnetic flux (Subscript[ϕ, B]) through the plane, degeneracy is

(Subscript[N, ϕ]):=((Subscript[ϕ, B])/(Subscript[ϕ, 0]))=((B A)/(Subscript[ϕ, 0])),

where

A - total area of the system,

(Subscript[ϕ, 0])=h/e - the magnetic flux quantum.

The key feature of electrons is their fermionic statistics. 
=> Pauli exclusion principle: no two electrons can occupy the same quantum state.

Filling up energy levels: Due to Pauli exclusion, electrons will fill up available quantum states starting from the lowest energy states.

Filling fraction ν is the (fractional) number of Landau levels that will be filled up,

N - total number of electron in the system,

n=N/A - electron density, i.e. the number of electrons per area.

Conversely, Eq. () allows us to express n in terms of ν,

n=((ν e)/h)B.

(Subscript[σ, H])=((n e)/B)=((ν (e^2))/h),

the lowest ν Landau levels are completely filled,

and all higher levels are completely empty.

Charge Pumping

Manipulate[Block[{a=0.3,M=9},Graphics[{{Lighter[Green,0.9],Annulus[{0,0},{a,Sqrt[M+1/2]}]},{Green,Opacity[ϕ],Circle[{0,0},0.5a],Disk[{0,0},0.2a],Text["\!\(ϕ\)",{0,-a},{0,0.8}]},{AbsoluteThickness[1],LightGray,Dashed,Table[Circle[{0,0},Sqrt[m]],{m,1,M}]},{Table[Circle[{0,0},Max[Sqrt[m-ϕ],a]],{m,1,M}]},{Opacity[1-Sqrt[1-ϕ]/a],Text["\!\( e\)",{a,0},{-1,0}],Text["\!\( -e\)",{-Sqrt[M+1/2],0},{0.5,0}]}},ImageSize->160]],{{ϕ,1,Style["\!\(ϕ/ϕ\_0\)","InlineFormula"]},0,1,ImageSize->Small},Paneled->False]

In addition to a uniform perpendicular magnetic field B, we thread an extra magnetic flux ϕ(t) through the central hole, which slowly increases from 0 to (Subscript[ϕ, 0]) over a long time T,

ϕ(t)=(Subscript[ϕ, 0])(t/T),

(Subscript[ϕ, 0])=h/e - flux quantum,

T≫1/(Subscript[ω, c]) - adiabatic evolution time. (t:0→T).

Due to the Aharonov-Bohm effect, each quantum state (Ket[n,m]) will be affected — its wave function radius (Subscript[r, m]) will be deformed adiabatically,

(Subscript[r, m])=(Sqrt[((Subscript[ϕ, 0])/(π B))])(Sqrt[m-t/T]).

Derive Eq. ().

(Subscript[ϕ, total])=π (Subscript[r, m]^2)B+ϕ(t),

((e (Subscript[ϕ, total]))/ℏ)=(e/ℏ)(π (Subscript[r, m]^2)B+ϕ(t))=2π m,

(Subscript[r, m]^2)=(1/(π B))(((2π ℏ m)/e)-ϕ(t))
=(1/(π B))((h/e)m-(Subscript[ϕ, 0])(t/T)),
=((Subscript[ϕ, 0])/(π B))(m-t/T).

the radius of every orbit shrinks from (Sqrt[m]) to (Sqrt[m-1]) => pumping one charge e from the outer edge to the inner edge.

During this process,

the radial current density:

(Subscript[j, r])=(1/(2π r))(e/T),

the circular electric field induced by the changing flux:

(Subscript[E, θ])=(1/(2π r))((dϕ(t))/(dt))=(1/(2π r))((Subscript[ϕ, 0])/T),

Hall conductivity for each fully filled Landau level:

(Subscript[σ, H])=((Subscript[j, r])/(Subscript[E, θ]))=(e/(Subscript[ϕ, 0]))=((e^2)/h).

Linear Response Theory

Current density operator for each electron (assuming m=e=h=1) reads

Justify Eq. () based on Eq. ().

j=n e v
=((N e)/(A m)) π
=N(B/(Subscript[N, ϕ]))((e^2)/(m h)) π,

π=m v is the kinetic momentum,

n=N/A is the electron density (number of electrons per area),

A is the total area of the system, related to the Landau level degeneracy (Subscript[N, ϕ]) by Eq. ()

(Subscript[N, ϕ])=((B A)/(Subscript[ϕ, 0]))=>A=(((Subscript[N, ϕ])(Subscript[ϕ, 0]))/B)=(((Subscript[N, ϕ])h)/(e B)).

(j/N)=(B/(Subscript[N, ϕ]))((e^2)/(m h)) π.

(OverHat[j])=(B/(Subscript[N, ϕ])) (OverHat[π]).

The expectation value of current density in the system is given by

⟨j⟩=(∑_(n,m ∈ occ))(Bra[n,m])(OverHat[j])(Ket[n,m])
=(B/(Subscript[N, ϕ]))(∑_(n,m ∈ occ))(Bra[n,m])(OverHat[π])(Ket[n,m]),

where (∑_(n,m ∈ occ)) is to sum over all lowest (Ket[n,m]) states that are occupied by the electron.

Based on Eq. (), the kinetic momentum operator (OverHat[π]) can be expressed as

As expected, the current density ⟨j⟩=0 vanishes on the many-body ground state, in the absence of perturbation.

Applying a (weak) electric field E to the system amounts to perturbing the vector potential A by a time-dependent perturbation δ,A(t),

A→A+δ,A(t),

with E(t)=-(∂_t)δ,A(t), according to Eq. ().

Assume that E(t)=E (e^(-i (ω+i (SubPlus[0]))t)) takes an oscillatory form with frequency ω, and is adiabatically turned on from the infinite past, then

δ,A(t)=(∫_(-∞)^t)dt'E(t')=-(E/(i ω))(e^(-i (ω+i (SubPlus[0])) t)).

Block[{ω=2π,ε=0.05},
Plot[Re[Exp[-I (ω+I ε) t]],{t,-50,2},PlotStyle->Black,PlotRange->{-1.2,1.2},FrameLabel->{"\!\(t/T\)","\!\(E\^\(-I(ω+I ε)t\)\)"},PlotLabel->StringTemplate["\!\(ε T=`2`\)"][ω,ε],AspectRatio->0.4,ImageSize->240]]

This means the Hamiltonian operator will be perturbed by

(OverHat[H])→(OverHat[H])+δ,(OverHat[H])(t),

with the perturbation given by

δ,(OverHat[H])(t)=((∂(OverHat[H]))/(∂A))·δ,A(t)

to the leading order of δ,A(t).

Given that (OverHat[π])=(OverHat[p])-A and

(OverHat[H])=(1/2)((OverHat[π])^2)=(1/2)((((OverHat[p])-A))^2),

we have

((∂(OverHat[H]))/(∂A))=-((OverHat[p])-A)=-(OverHat[π]).

Substitute Eq. () into Eq. (), the time-dependent perturbation Hamiltonian reads

Dressed Green’s function (unitary time-evolution operator) can be computed from the Dyson series to the leading order

where the bare Green’s function is defined by

with (Subscript[E, n])=B (n+1/2) being the Landau level energy (see Eq. ()).

Every state will evolve in time by

(Ket[n,m])→(OverHat[G])(t,-∞)(Ket[n,m]).

As a result, by Eq. (), the current density expectation value ⟨j⟩ evolves as

⟨j(t)⟩=(B/(Subscript[N, ϕ]))(∑_(n,m ∈ occ))(Bra[n,m])(OverHat[G])(-∞,t)(OverHat[π])(OverHat[G])(t,-∞)(Ket[n,m])
=(B/(Subscript[N, ϕ]))(∑_(n,m ∈ occ))(Bra[n,m])((OverHat[π])(t)+(1/ω)(∫_(-∞)^t)dt'(e^(-i (ω+i (SubPlus[0])) t'))[E·(OverHat[π])(t'),(OverHat[π])(t)]+…)(Ket[n,m]).

Derive Eq. ().

⟨j(t)⟩=(B/(Subscript[N, ϕ]))(∑_(n,m ∈ occ))(Bra[n,m])(OverHat[G])(-∞,t)(OverHat[π])(OverHat[G])(t,-∞)(Ket[n,m])
=(B/(Subscript[N, ϕ]))(∑_(n,m ∈ occ))(Bra[n,m])((Subscript[(OverHat[G]), 0])(-∞,t)(OverHat[π])(OverHat[(Subscript[G, 0])])(t,-∞)+(-i)(∫_(-∞)^t)dt'(Subscript[(OverHat[G]), 0])(-∞,t)(OverHat[π])(Subscript[(OverHat[G]), 0])(t,t')δ(OverHat[H])(t')(Subscript[(OverHat[G]), 0])(t',-∞)-(-i)(∫_(-∞)^t)dt'(Subscript[(OverHat[G]), 0])(-∞,t')δ(OverHat[H])(t')(Subscript[(OverHat[G]), 0])(t',t)(OverHat[π])(OverHat[(Subscript[G, 0])])(t,-∞)+…)(Ket[n,m]).

(OverHat[π])(t):=(Subscript[(OverHat[G]), 0])(-∞,t)(OverHat[π])(OverHat[(Subscript[G, 0])])(t,-∞),

δ,(Subscript[(OverHat[H]), ℐ])(t):=(Subscript[(OverHat[G]), 0])(-∞,t)δ,(OverHat[H])(t)(OverHat[(Subscript[G, 0])])(t,-∞)
=(Subscript[(OverHat[G]), 0])(-∞,t)(1/(i ω))E·(OverHat[π]) (e^(-i (ω+i (SubPlus[0])) t))(OverHat[(Subscript[G, 0])])(t,-∞)
=(1/(i ω))E·(OverHat[π])(t) (e^(-i (ω+i (SubPlus[0])) t)).

⟨j(t)⟩=(B/(Subscript[N, ϕ]))(∑_(n,m ∈ occ))(Bra[n,m])((OverHat[π])(t)+(-i)(∫_(-∞)^t)dt'(OverHat[π])(t)δ(Subscript[(OverHat[H]), ℐ])(t')-(-i)(∫_(-∞)^t)dt'δ(Subscript[(OverHat[H]), ℐ])(t')(OverHat[π])(t)+…)(Ket[n,m])
=(B/(Subscript[N, ϕ]))(∑_(n,m ∈ occ))(Bra[n,m])((OverHat[π])(t)+i(∫_(-∞)^t)dt'[δ(Subscript[(OverHat[H]), ℐ])(t'),(OverHat[π])(t)]+…)(Ket[n,m])
=(B/(Subscript[N, ϕ]))(∑_(n,m ∈ occ))(Bra[n,m])((OverHat[π])(t)+(1/ω)(∫_(-∞)^t)dt'(e^(-i (ω+i (SubPlus[0])) t'))[E·(OverHat[π])(t'),(OverHat[π])(t)]+…)(Ket[n,m]).

where we have introduced

(OverHat[π])(t):=(Subscript[(OverHat[G]), 0])(-∞,t)(OverHat[π])(OverHat[(Subscript[G, 0])])(t,-∞).

The first term in Eq. () is the current density in the absence of an electric field, which vanishes.

(∑_(n,m ∈ occ))(Bra[n,m])(OverHat[π])(t)(Ket[n,m])=0.

Show Eq. ().

(Bra[n,m])(OverHat[π])(t)(Ket[n,m])
=(Bra[n,m])(Subscript[(OverHat[G]), 0])(-∞,t)(OverHat[π])(OverHat[(Subscript[G, 0])])(t,-∞)(Ket[n,m])
=(Bra[n,m])(e^(i (Subscript[E, n])(t+∞)))(OverHat[π])(e^(-i (Subscript[E, n])(t+∞)))(Ket[n,m])
=(Bra[n,m])(OverHat[π])(Ket[n,m]).

(Bra[n,m])(Subscript[(OverHat[π]), x])(Ket[n,m])=(Sqrt[(B/2)])(Bra[n,m])((SuperDagger[(OverHat[a])])+(OverHat[a]))(Ket[n,m])
=(Sqrt[(B/2)])((Sqrt[n+1])(BraKet[n,m, n+1,m])+(Sqrt[n])(BraKet[n,m, n-1,m]))
=0,
(Bra[n,m])(Subscript[(OverHat[π]), y])(Ket[n,m])=(Sqrt[(B/2)])i (Bra[n,m])((SuperDagger[(OverHat[a])])-(OverHat[a]))(Ket[n,m])
=(Sqrt[(B/2)])i ((Sqrt[n+1])(BraKet[n,m, n+1,m])-(Sqrt[n])(BraKet[n,m, n-1,m]))
=0,

The second term in Eq. () describes the linear response of the current density under the electric field, which takes the form of

with the conductivity matrix (Subscript[σ, j i])(ω) given by

(Subscript[σ, j i])(ω)=(B/(ω (Subscript[N, ϕ])))(∫_(-∞)^0)dt (e^(-i (ω+i (SubPlus[0])) t))(∑_(n,m ∈ occ))(Bra[n,m])[(Subscript[(OverHat[π]), j])(t),(Subscript[(OverHat[π]), i])(0)](Ket[n,m]).

Derive Eq. () and Eq. ().

⟨j(t)⟩=(B/(ω (Subscript[N, ϕ])))(∫_(-∞)^t)dt'(e^(-i (ω+i (SubPlus[0])) t'))(∑_(n,m ∈ occ))(Bra[n,m])[E·(OverHat[π])(t'),(OverHat[π])(t)](Ket[n,m])
=((B (e^(-i (ω+i (SubPlus[0])) t)))/(ω (Subscript[N, ϕ])))(∫_(-∞)^t)dt'(e^(-i (ω+i (SubPlus[0])) (t'-t)))(∑_(n,m ∈ occ))(Bra[n,m])[E·(OverHat[π])(t'),(OverHat[π])(t)](Ket[n,m]).

⟨j(t)⟩
=((B (e^(-i (ω+i (SubPlus[0])) t)))/(ω (Subscript[N, ϕ])))(∫_(-∞)^0)dt''(e^(-i (ω+i (SubPlus[0])) t''))(∑_(n,m ∈ occ))(Bra[n,m])[E·(OverHat[π])(t''),(OverHat[π])(0)](Ket[n,m]).

(Subscript[σ, j i])(ω)=(B/(ω (Subscript[N, ϕ])))(∫_(-∞)^0)dt (e^(-i (ω+i (SubPlus[0])) t))(∑_(n,m ∈ occ))(Bra[n,m])[(Subscript[(OverHat[π]), j])(t),(Subscript[(OverHat[π]), i])(0)](Ket[n,m]),

⟨(Subscript[j, i])(t)⟩=(Subscript[E, j])(Subscript[σ, j i])(ω)(e^(-i (ω+i (SubPlus[0])) t)).

Eq. () implies that when the applied electric field E(t)=E (e^(-i (ω+i (SubPlus[0]))t)) oscillates at frequency ω, the induced current ⟨j(t)⟩ also oscillates at the same frequency. — A feature of linear response.

The Hall conductivity corresponds to the off-diagonal component (Subscript[σ, x y])(ω) of the conductivity matrix. In the DC limit ω->0, it is given by the Kobo formula:

Derive Eq. ().

(Bra[n,m])[(Subscript[(OverHat[π]), j])(t),(Subscript[(OverHat[π]), i])(0)](Ket[n,m])
=(∑_(n',m'))((Bra[n,m])(Subscript[(OverHat[π]), j])(t)(Ket[n',m'])(Bra[n',m'])(Subscript[(OverHat[π]), i])(0)(Ket[n,m])-(Bra[n,m])(Subscript[(OverHat[π]), i])(0)(Ket[n',m'])(Bra[n',m'])(Subscript[(OverHat[π]), j])(t)(Ket[n,m]))
=(∑_(n',m'))((Bra[n,m])(e^(i (Subscript[E, n])t))(Subscript[(OverHat[π]), j])(e^(-i (Subscript[E, n'])t))(Ket[n',m'])(Bra[n',m'])(Subscript[(OverHat[π]), i])(Ket[n,m])-(Bra[n,m])(Subscript[(OverHat[π]), i])(Ket[n',m'])(Bra[n',m'])(e^(i (Subscript[E, n'])t))(Subscript[(OverHat[π]), j])(e^(-i (Subscript[E, n])t))(Ket[n,m]))
=(∑_(n',m'))((Bra[n,m])(Subscript[(OverHat[π]), j])(Ket[n',m'])(Bra[n',m'])(Subscript[(OverHat[π]), i])(Ket[n,m])(e^(i ((Subscript[E, n])-(Subscript[E, n']))t))-(Bra[n,m])(Subscript[(OverHat[π]), i])(Ket[n',m'])(Bra[n',m'])(Subscript[(OverHat[π]), j])(Ket[n,m])(e^(-i ((Subscript[E, n])-(Subscript[E, n']))t))),

(Subscript[σ, j i])(ω)=(B/(ω (Subscript[N, ϕ])))(∫_(-∞)^0)dt (∑_(n,m ∈ occ))(∑_(n',m'))((Bra[n,m])(Subscript[(OverHat[π]), j])(Ket[n',m'])(Bra[n',m'])(Subscript[(OverHat[π]), i])(Ket[n,m])(e^(-i (ω+i (SubPlus[0])-(Subscript[E, n])+(Subscript[E, n'])) t))-(Bra[n,m])(Subscript[(OverHat[π]), i])(Ket[n',m'])(Bra[n',m'])(Subscript[(OverHat[π]), j])(Ket[n,m])(e^(-i (ω+i (SubPlus[0])+(Subscript[E, n])-(Subscript[E, n'])) t)))
=((i B)/(ω (Subscript[N, ϕ])))(∑_(n,m ∈ occ))(∑_(n',m'))((((Bra[n,m])(Subscript[(OverHat[π]), j])(Ket[n',m'])(Bra[n',m'])(Subscript[(OverHat[π]), i])(Ket[n,m]))/(ω-(Subscript[E, n])+(Subscript[E, n'])))-(((Bra[n,m])(Subscript[(OverHat[π]), i])(Ket[n',m'])(Bra[n',m'])(Subscript[(OverHat[π]), j])(Ket[n,m]))/(ω+(Subscript[E, n])-(Subscript[E, n'])))).

(Subscript[(OverHat[π]), i])(Ket[n,m])→(Ket[n±1,m]),

(Subscript[σ, j i])(ω)=((i B)/(ω (Subscript[N, ϕ])))(∑_(n,m ∈ occ))(∑_(n'))((((Bra[n,m])(Subscript[(OverHat[π]), j])(Ket[n',m])(Bra[n',m])(Subscript[(OverHat[π]), i])(Ket[n,m]))/(ω-(Subscript[E, n])+(Subscript[E, n'])))-(((Bra[n,m])(Subscript[(OverHat[π]), i])(Ket[n',m])(Bra[n',m])(Subscript[(OverHat[π]), j])(Ket[n,m]))/(ω+(Subscript[E, n])-(Subscript[E, n'])))).

(1/(ω-(Subscript[E, n])+(Subscript[E, n'])))=(1/((Subscript[E, n'])-(Subscript[E, n])))-(ω/((((Subscript[E, n'])-(Subscript[E, n])))^2))+…,
(1/(ω+(Subscript[E, n])-(Subscript[E, n'])))=-(1/((Subscript[E, n'])-(Subscript[E, n])))-(ω/((((Subscript[E, n'])-(Subscript[E, n])))^2))+…,

(Subscript[σ, j i])(ω)=((i B)/(ω (Subscript[N, ϕ])))(∑_(n,m ∈ occ))(∑_(n'))(1/((Subscript[E, n'])-(Subscript[E, n])))((Bra[n,m])(Subscript[(OverHat[π]), j])(Ket[n',m])(Bra[n',m])(Subscript[(OverHat[π]), i])(Ket[n,m])+h.c.)
-((i B)/(Subscript[N, ϕ]))(∑_(n,m ∈ occ))(∑_(n'))(1/((((Subscript[E, n'])-(Subscript[E, n])))^2))((Bra[n,m])(Subscript[(OverHat[π]), j])(Ket[n',m])(Bra[n',m])(Subscript[(OverHat[π]), i])(Ket[n,m])-h.c.)

The first term looks divergent. Indeed, such divergences do arise for longitudinal conductivities. However, it does not contribute to the Hall conductivity.

The Hall conductivity only receives contribution from the second term,

(Subscript[σ, x y])(ω)=(B/(i (Subscript[N, ϕ])))(∑_(n,m ∈ occ))(∑_(n'))(((Bra[n,m])(Subscript[(OverHat[π]), x])(Ket[n',m])(Bra[n',m])(Subscript[(OverHat[π]), y])(Ket[n,m])-h.c.)/((((Subscript[E, n'])-(Subscript[E, n])))^2)).

Using Eq. () to represent the kinetic momentum operator (OverHat[π]) as annihilation and creation operators, and given that

(OverHat[a])(Ket[n,m])=(Sqrt[n])(Ket[n-1,m]),
(SuperDagger[(OverHat[a])])(Ket[n,m])=(Sqrt[n+1])(Ket[n+1,m]),

the Hall conductivity (Subscript[σ, H]) in Eq. () reduces to

Derive Eq. ().

Substitute Eq. () into Eq. (),

(Bra[n,m])(Subscript[(OverHat[π]), x])(Ket[n',m])(Bra[n',m])(Subscript[(OverHat[π]), y])(Ket[n,m])
=((i B)/2)(Bra[n,m])((SuperDagger[(OverHat[a])])+(OverHat[a]))(Ket[n',m])(Bra[n',m])((SuperDagger[(OverHat[a])])-(OverHat[a]))(Ket[n,m])
=((i B)/2)((Bra[n,m])(OverHat[a])(Ket[n',m])(Bra[n',m])(SuperDagger[(OverHat[a])])(Ket[n,m])-(Bra[n,m])(SuperDagger[(OverHat[a])])(Ket[n',m])(Bra[n',m])(OverHat[a])(Ket[n,m]))
=((i B)/2)((n+1)((|(BraKet[n',m, n+1,m])|)^2)-n((|(BraKet[n',m, n-1,m])|)^2))
=((i B)/2)((n+1)(Subscript[δ, n',n+1])-n (Subscript[δ, n',n-1])).

(Subscript[σ, H])=(B/(i (Subscript[N, ϕ])))(∑_(n,m ∈ occ))(∑_(n'≠n))(1/((((Subscript[E, n'])-(Subscript[E, n])))^2))(((i B)/2)((n+1)(Subscript[δ, n',n+1])-n (Subscript[δ, n',n-1]))-h.c.)
=(B/(i (Subscript[N, ϕ])))(∑_(n,m ∈ occ))(∑_(n'≠n))((i B)/((((Subscript[E, n'])-(Subscript[E, n])))^2))((n+1)(Subscript[δ, n',n+1])-n (Subscript[δ, n',n-1]))
=((B^2)/(Subscript[N, ϕ]))(∑_(n,m ∈ occ))(∑_(n'≠n))((((n+1)(Subscript[δ, n',n+1]))/((((Subscript[E, n'])-(Subscript[E, n])))^2))-((n (Subscript[δ, n',n-1]))/((((Subscript[E, n'])-(Subscript[E, n])))^2)))
=((B^2)/(Subscript[N, ϕ]))(∑_(n,m ∈ occ))(((n+1)/((((Subscript[E, n+1])-(Subscript[E, n])))^2))-(n/((((Subscript[E, n-1])-(Subscript[E, n])))^2))).

(Subscript[σ, H])=((B^2)/(Subscript[N, ϕ]))(∑_(n,m ∈ occ))(((n+1)/(B^2))-(n/(B^2)))
=(1/(Subscript[N, ϕ]))(∑_(n,m ∈ occ))1.

Each occupied state contributes 1/(Subscript[N, ϕ]) to (Subscript[σ, H]) (in unit of the conductance quantum (e^2)/h).

Each Landau level is (Subscript[N, ϕ])-fold degenerated. => Fully filling each Landau level produces exactly one unit of (Subscript[σ, H]).

Spin and Monopole

Classical Spin

Angular Momentum Decomposition

τ=((dL)/(dt)),

τ - torque exerted on the top,

L - total angular momentum of the top, decomposed into components parallel (Subscript[I, a])ω and perpendicular (Subscript[I, t])Ω to the spinning axis

L=(Subscript[I, a])ω+(Subscript[I, t])Ω,

Block[{r=1,d=0.05,h=1,a=0.05,θ=π/8,ϕ=-π/3,α=π/24,th=0.016,c=0.2,viewpt,dashedtube,rot},
viewpt={1.3, -2.4, 1.5};
dashedtube[v_]:=Tube[Table[{x v,(x+0.05)v},{x,-1,0.9,0.1}],th];
rot[g_]:=Rotate[Rotate[g,θ,{1,0,0},{0,0,0}],ϕ,{0,0,1},{0,0,0}];
Graphics3D[{Translate[rot[{White,AbsoluteThickness[5],Line[# h{0,0,1}&/@{-1.3,-1.4}],
Line[# r{0,1,0}&/@{-1.15,-1.2}]}],-0.02viewpt],rot[{{EdgeForm[],Specularity[White,5],Gray,Tube[{0,0,#}&/@{-h,h},a],Opacity[0.8],Yellow,Cylinder[{0,0,#}&/@{-d,d},r]},{Black,dashedtube[1.7h{0,0,1}],dashedtube[1.5r{1,0,0}],dashedtube[1.5r{0,1,0}]},{Black,Tube[Table[{0,0,-1.5h}+c{Cos[β],Sin[β],0},{β,0,2π,π/32}],th],Text["\!\(I\_a\)",{0,0,-1.5h},{2,0.8}],Tube[Table[{-1.2r,0,0}+c{0,Cos[β],Sin[β]},{β,0,2π,π/32}],th],Text["\!\(I\_t\)",{-1.2r,0,0},{-1.2,-1.2}],Tube[Table[{0,-1.2r,0}+c{Cos[β],0,Sin[β]},{β,0,2π,π/32}],th],Text["\!\(I\_t\)",{0,-1.2r,0},{1,1.5}]}}],Translate[rot[{Arrowheads[0.05],Block[{vL=2{0,Sin[α],Cos[α]},vS=2{0,0,Cos[α]}},{Blue,Arrow[# vL&/@{0,1}],Text["\!\( L\)",vL,{-2,1.5}],LightBlue,Arrow[vS#&/@{0,1}],Text["\!\( S\)",vS,{1,1}]}],{Red,Arrow[{0,0,1.5h}#&/@{0,1}],Text["\!\(ω\)",{0,0,1.5h},{1,0.5}],Arrow[{0,0.4r,0}#&/@{0,1}],Text["\!\(Ω\)",0.4r{0,1,0},{0,1}]}}],0.04viewpt]},Boxed->False,Lighting->"Neutral",ViewPoint->viewpt,ImageSize->140]]

(Subscript[I, a]) - axial moment of inertia (about the spinning axis),

(Subscript[I, t]) - transverse moment of inertia (about either of the two equivalent transverse axes)

ω - axial angular velocity (along the spinning axis)

Ω - transverse angular velocity, describing the instantaneous rotation rate of the spinning axis itself.

Dynamics of Spinning Axis

τ=(Subscript[I, a])((dω)/(dt))+(Subscript[I, t])((dΩ)/(dt)).

Transverse Torque Assumption: Assume τ has no component along the spinning axis, so the magnitude of ω remains constant. Only its direction changes due to the rotation of the spinning axis:

((dω)/(dt))=Ω×ω,

Eq. () becomes

τ=(Subscript[I, a])Ω×ω+(Subscript[I, t])((dΩ)/(dt)).

((dn)/(dt))=Ω×n,

Ω=n×((dn)/(dt)).

Derive Eq. () from Eq. ().

n×((dn)/(dt))=n×(Ω×n)=Ω (n·n)-n (n·Ω)=Ω,

S:=(Subscript[I, a])ω=S n - spin angular momentum (parallel to the spinning axis),

(Subscript[((Overscript[n, ¨])), ⊥])=(Overscript[n, ¨])-((Overscript[n, ¨])·n) n  - component of acceleration (Overscript[n, ¨]) in the tangent plane.

Derive Eq. ().

τ=(Subscript[I, a])(n×((dn)/(dt)))×ω+(Subscript[I, t])(d/(dt))(n×((dn)/(dt)))
=(Subscript[I, a])(n·ω)((dn)/(dt))-(Subscript[I, a])n(((dn)/(dt))·ω)+(Subscript[I, t])((dn)/(dt))×((dn)/(dt))+(Subscript[I, t])n×(((d^2)n)/(d(t^2))),

(dn/dt)·ω=0, because the normalization condition n·n=1 of the unit vector n implies

2((dn)/(dt))·n=(d/(dt))(n·n)=(d/(dt))(1)=0,

therefore,

((dn)/(dt))·ω=ω((dn)/(dt))·n=0.

(dn/dt)×(dn/dt)=0, because any vector cross product with itself always vanishes.

τ=(Subscript[I, a]) ω((dn)/(dt))+(Subscript[I, t])n×(((d^2)n)/(d(t^2)))
=S((dn)/(dt))+(Subscript[I, t])n×(((d^2)n)/(d(t^2))),

τ×n=S((dn)/(dt))×n+(Subscript[I, t]) (n×(((d^2)n)/(d(t^2))))×n,
=((dn)/(dt))×S+(Subscript[I, t]) ((n·n)(((d^2)n)/(d(t^2)))-n ((((d^2)n)/(d(t^2)))·n)),

(n×(((d^2)n)/(d(t^2))))×n=(n·n)(((d^2)n)/(d(t^2)))-n ((((d^2)n)/(d(t^2)))·n)
=(((d^2)n)/(d(t^2)))-n ((((d^2)n)/(d(t^2)))·n)
=(Subscript[((((d^2)n)/(d(t^2)))), ⊥])

(Subscript[I, t]) (Subscript[((((d^2)n)/(d(t^2)))), ⊥])=τ×n-((dn)/(dt))×S.

Magnetic Monopole

Electromagnetic Analogy

Analogy: Compare the spin dynamics in Eq. () and the charge dynamics in Eq. ()

({{spin:, (Subscript[I, t]) (Subscript[((Overscript[n, ¨])), ⊥]), =, τ×n, -, (OverDot[n])×S}, {charge:, m (Overscript[x, ¨]), =, E, +, (OverDot[x])×B}})

({{Spin dynamics, Charge dynamics}, {Spin orientation:n, Charge position:x}, {Moment of inertia:(Subscript[I, t]), Mass (interia):m}, {Torque-induced force:τ×n, Electric field:E}, {Spin angular momentum:-S, Magnetic field:B}})

Similarity: Just as in electromagnetism, where the Lorentz force deflects a charge moving in a magnetic field, the spin-induced term -(OverDot[n])×S generates precession of the spinning axis.

Block[{r=0.8,d=0.05,h=1,a=0.04,θ=π/6,ϕ=-π/6,th=0.02,c=0.08,viewpt,dashedtube,rot},
viewpt={0, -2.4,0.8};
dashedtube[v_]:=Tube[Table[{x v,(x+0.05)v},{x,0,0.9,0.1}],th];
rot[g_]:=Rotate[Rotate[g,θ,{1,0,0},{0,0,0}],ϕ,{0,0,1},{0,0,0}];
Graphics3D[{rot[Translate[{{EdgeForm[],Specularity[White,5],Gray,Tube[{0,0,#}&/@{-h,h},a],Opacity[0.8],Orange,Cylinder[{0,0,#}&/@{-d,d},r]},{Red,Sphere[{0,0,h},c]},(*{Black,Sphere[{0,0,-h},c]},*)Translate[{Arrowheads[0.04],{Blue,Arrow[Tube[{{0,0,0},{0,0,1}},th]],Text["\!\( -B=S\)",{0,0,1},{1.1,1}]},{Red,Arrow[Tube[{{0,0,0},{1,0,0}},th]],Text["\!\(x\&.=n\&.\)",{1,0,0},{-0.7,1}]},{Purple,Arrow[Tube[{{0,0,0},{0,1,0}},th]],Text["\!\(x\&.×B=-n\&.×S\)",{0,1,0},{-0.7,-0.5}]},{Green,Arrow[Tube[{{0,0,0},{0,-1,0}},th]],Text["\!\( E=τ×n\)",{0,-1,0},{1,-0.3}]}},{0,0,h}]},{0,0,h}]],{Opacity[0.1],Sphere[{0,0,0},2h]},{{Glow[Gray],Black,dashedtube[{0,0,2h}],dashedtube[{0,0,-2h}],dashedtube[2h{Sin[ϕ],-Cos[ϕ],0}],dashedtube[{2h,0,0}],dashedtube[{-2h,0,0}],Tube[2h{Cos[#],Sin[#],0}&/@Range[0,2π,π/32],th],
Tube[2h{Cos[#]Sin[θ],Sin[#]Sin[θ],Cos[θ]}&/@Range[0,2π,π/32],th],Tube[2h{Sin[ϕ]Sin[#],-Cos[ϕ]Sin[#],Cos[#]}&/@Range[0,π,π/32],th],Block[{β=π/2},Tube[2h{Sin[β]Sin[#],-Cos[β]Sin[#],Cos[#]}&/@Range[0,2π,π/32],th]]}}},Boxed->False,Lighting->"Neutral",ViewPoint->viewpt,ImageSize->{Automatic,160}]]

Difference: The constraint to the sphere makes the coordinate system non-Euclidean (curved), in which ∇ operator is defined differently. []

Block[{r=0.8,d=0.05,h=1,a=0.04,θ=π/5,ϕ=-π/6,th=0.02,c=0.4,viewpt,dashedtube,rot},
viewpt={0, -2.4,0.8};
dashedtube[v_,sp_:0.1]:=Tube[Table[{x v,(x+0.05)v},{x,0,0.9,sp}],th];
rot[g_]:=Rotate[Rotate[g,θ,{1,0,0},{0,0,0}],ϕ,{0,0,1},{0,0,0}];
Row[{Spacer[25],Graphics3D[{rot[Translate[{Arrowheads[0.04],{Black,Arrow[Tube[{{0,0,-h},{0,0,h}},th]],Text["\!\( n\)",{0,0,h},{0.7,-0.55}]},Translate[{Black,Arrow[Tube[{{0,0,0},{1,0,0}},th]],Text["\!\(e\_φ\)",{1,0,0},{-0.,0.8}],Arrow[Tube[{{0,0,0},{0,-1,0}},th]],Text["\!\(e\_θ\)",{0,-1,0},{1.2,-0.5}]},{0,0,h}]},{0,0,h}]],{Opacity[0.1],Sphere[{0,0,0},2h]},{{Glow[Gray],Black,dashedtube[{0,0,2h}],dashedtube[{0,0,-2h}],Translate[dashedtube[2h Sin[θ]{Sin[ϕ],-Cos[ϕ],0},0.2],2h{0,0,Cos[θ]}],dashedtube[2h{Sin[ϕ],-Cos[ϕ],0}],dashedtube[{2h,0,0}],dashedtube[{-2h,0,0}],Tube[2h{Cos[#],Sin[#],0}&/@Range[0,2π,π/32],th],
Tube[2h{Cos[#]Sin[θ],Sin[#]Sin[θ],Cos[θ]}&/@Range[0,2π,π/32],th],Tube[2h{Sin[ϕ]Sin[#],-Cos[ϕ]Sin[#],Cos[#]}&/@Range[0,π,π/32],th],
Block[{β=π/2},Tube[2h{Sin[β]Sin[#],-Cos[β]Sin[#],Cos[#]}&/@Range[0,2π,π/32],th]]},{Glow[Black],Black,Tube[c{Sin[ϕ]Sin[#],-Cos[ϕ]Sin[#],Cos[#]}&/@Range[0,θ,π/32],th],Text["\!\(θ\)",1.6c{Sin[ϕ]Sin[θ/2],-Cos[ϕ]Sin[θ/2],Cos[θ/2]}],Tube[c{-Cos[#],-Sin[#],0}&/@Range[0,π/2+ϕ,π/32],th],Text["\!\(φ\)",-1.6c{Cos[π/4+ϕ/2],Sin[π/4+ϕ/2],0}]}}},Boxed->False,Lighting->"Neutral",ViewPoint->viewpt,ImageSize->{Automatic,136}]}]]

Spherical coordinate: parametrize the spin axis

n=(cos φ sin θ, sin φ sin θ, cos θ),

by the polar angle θ∈[0,π] and the azimuthal angle φ∈[0,2π).

Unit vectors:

(e^θ)=(cos φ cos θ,sin φ cos θ,-sin θ),
(e^φ)=(-sin φ,cos φ,0).

Surface gradient: (Subscript[∇, ⊥])Φ for a scalar field Φ,

(Subscript[∇, ⊥])Φ=(e^θ)(∂_θ)Φ+(e^φ)(1/(sin θ))(∂_φ)Φ.

Surface curl: (Subscript[∇, ⊥])×A for a vector field A=(Subscript[A, θ])(e^θ)+(Subscript[A, φ])(e^φ),

(Subscript[∇, ⊥])×A=(1/(sin θ))((∂_θ)((Subscript[A, φ])sin θ)-(∂_φ)(Subscript[A, θ]))n.

Wikipedia. Del in cylindrical and spherical coordinates.

Differential Geometry for Vector Calculus

Differential Forms: differential form is the basic object in differential geometry, it is an object that you can integrate:

A 0-form is just a scalar field f(x), which can be integrated (evaluated) on a point x.

A 1-form is a vector field (line element), like ω(x)=(Subscript[ω, i])(x)d(x^i), something you can integrate along a curve.

A 2-form is an tensor field (oriented surface element), e.g. (Subscript[σ, i j])(x)d(x^i)⋀d(x^j), integrable over a surface.

In general:

A k-form is something you integrate over a k-dimensional submanifold.

The wedge product ⋀ defines an oriented, antisymmetric product between forms:

d(x^i)⋀d(x^j)=-d(x^j)⋀d(x^i).

Metric: The metric defines how distance is measured on the manifold.

where

d(s^2) is the squared infinitesimal distance element,

(Subscript[g, i j])(x) are the components of the metric tensor, forming a symmetric positive-definite matrix.

d(x^i) are the differential 1-form basis (cotangent basis)

Exterior Derivative: the exterior derivative d acts on differential forms to produce forms of higher degree. Given f=(Subscript[f, I])d(x^I),

(d^2)=0: the exterior derivative of an exterior derivative is always zero.

Stoke’s Theorem:

Hodge Dual: On an n-dimensional oriented manifold, the Hodge star operator ★ maps k-forms to (n-k)-forms.

★(d(x^(Subscript[i, 1]))⋀…⋀d(x^(Subscript[i, k])))=(((|det[(Subscript[g, i j])]|)^(1/2))/((n-k)!))(g^((Subscript[i, 1])(Subscript[j, 1])))… (g^((Subscript[i, k])(Subscript[j, k])))(Subscript[ϵ, (Subscript[j, 1])… (Subscript[j, n])])d(x^(Subscript[j, k+1]))⋀…⋀d(x^(Subscript[j, n])),

where

(Subscript[ϵ, (Subscript[j, 1])… (Subscript[j, n])]) is the Levi-Civita symbol,

(g^(i j))=⟨d(x^i),d(x^j)⟩ is the inverse metric, which tells how differential forms d(x^i) and d(x^j)are “angled” with respect to each other.

Metric: the distance element is given by

d(s^2)=d(θ^2)+(sin^2)θ d(φ^2),

which implies

(Subscript[g, i j])=(({{1, 0}, {0, (sin^2)θ}})),(g^(i j))=(({{1, 0}, {0, (sin^(-2))θ}})),

and ((|det[(Subscript[g, i j])]|)^(1/2))=sin θ.

Unit (co)vectors: orthonormal basis of 1-forms

This enables us to represent any vector field as 1-form field, such as

A=(Subscript[A, θ])(e^θ)+(Subscript[A, φ])(e^φ)↔A=(Subscript[A, θ])dθ+(Subscript[A, φ])sin θ dφ.

Gradient: given a scalar field Φ,

Derive Eq. () using differential geometry approach.

grad Φ↔dΦ=dθ (∂_θ)Φ+dφ (∂_φ)Φ↔(e^θ) (∂_θ)Φ+(e^φ)(1/(sin θ))(∂_φ)Φ.

Curl: given a vector field A=(Subscript[A, θ])(e^θ)+(Subscript[A, φ])(e^φ),

Derive Eq. () using differential geometry approach.

curl A↔★dA=★d((Subscript[A, θ])dθ+(Subscript[A, φ])sin θ dφ)
=★((∂_φ)(Subscript[A, θ])dφ⋀dθ+(∂_θ)((Subscript[A, φ])sin θ)dθ⋀dφ)
=((∂_θ)((Subscript[A, φ])sin θ)-(∂_φ)(Subscript[A, θ]))★(dθ⋀dφ)
=(1/(sin θ))((∂_θ)((Subscript[A, φ])sin θ)-(∂_φ)(Subscript[A, θ])).

The result is a 0-form, indicating a scalar field on the sphere. But alternatively, we may as well view it as a vector field along the normal direction of the sphere,

curl A=(1/(sin θ))((∂_θ)((Subscript[A, φ])sin θ)-(∂_φ)(Subscript[A, θ]))n.

Divergence: given a vector field A=(Subscript[A, θ])(e^θ)+(Subscript[A, φ])(e^φ),

Derive Eq. () using differential geometry approach.

div A↔★d★ A=★d★((Subscript[A, θ])dθ+(Subscript[A, φ])sin θ dφ)
=★d((Subscript[A, θ]) (sin θdφ)+(Subscript[A, φ])sin θ (-(1/(sin θ))dθ))
=★d((Subscript[A, θ]) sin θdφ-(Subscript[A, φ])dθ)
=★((∂_θ)((Subscript[A, θ]) sin θ)dθ⋀dφ-(∂_φ)(Subscript[A, φ])dφ⋀dθ)
=((∂_θ)((Subscript[A, θ]) sin θ)+(∂_φ)(Subscript[A, φ]))★(dθ⋀dφ)
=(1/(sin θ))((∂_θ)((Subscript[A, θ]) sin θ)+(∂_φ)(Subscript[A, φ])).

Laplacian: given a scalar field ψ,

Derive Eq. () using differential geometry approach.

(∇^2)ψ↔★d★dψ
=★d★(dθ(∂_θ)ψ+dφ(∂_φ)ψ)
=★d((∂_θ)ψ (sin θdφ)+(∂_φ)ψ(-(1/(sin θ))dθ))
=★d(sin θ (∂_θ)ψ dφ-(1/(sin θ))(∂_φ)ψdθ)
=★((∂_θ)(sin θ (∂_θ)ψ)dθ⋀dφ-(1/(sin θ))(∂_φ^2)ψdφ⋀dθ)
=((∂_θ)(sin θ (∂_θ)ψ)+(1/(sin θ))(∂_φ^2)ψ)★(dθ⋀dφ)
=(1/(sin θ))((∂_θ)(sin θ (∂_θ)ψ)+(1/(sin θ))(∂_φ^2)ψ).

Vincent Bouchard. MATH 315: Calculus IV (University of Alberta).

Effective Gauge Field

Effective electric field:

In this way, E is guaranteed to lie in the tangent plane, the same as the torque-induced force τ×n.

Block[{dθ=π/8,dϕ=π/6,θs,ϕs,cfun},
θs=Range[dθ,π-dθ,dθ];
ϕs=Range[dϕ,2π,dϕ];
cfun=ColorData["RedBlueTones"];
Row[{Show[{SphericalPlot3D[1,{θ,0,π},{ϕ,0,2π},Mesh->{θs,ϕs},Axes->False,Boxed->False,ColorFunction->(cfun[#3]&),MeshStyle->Directive[Black,AbsoluteThickness[0.8],Opacity[0.4]],Lighting->"Neutral",ViewPoint->{0, -2.4, 0.8}],Graphics3D[{{Glow[Black],Black,Arrowheads[0.04],Function[{θ,ϕ},Translate[Arrow[Tube[{{0,0,0},0.35Sin[θ]{Cos[ϕ]Cos[θ],Sin[ϕ]Cos[θ],-Sin[θ]}},0.015]],{Cos[ϕ]Sin[θ],Sin[ϕ]Sin[θ],Cos[θ]}]]@@@Tuples[N@{θs,ϕs}]},{Black,Text[Style["\!\( E\)","Text"],{-1,0,0},{2,-0.5}]}},Lighting->"Neutral"]},PlotRange->All,ImageSize->125],Graphics[Raster[List/@Range[0,1,0.02],ColorFunction->cfun],PlotLabel->Style["\!\(Φ\)",Black,"Text"],AspectRatio->8,ImageSize->{Automatic,80}]}]]

Example: a spinning top in a uniform gravity field Φ(n)∝(Subscript[n, z])=cos θ,

E=-(Subscript[∇, ⊥])Φ=(Subscript[e, θ])sin θ.

Effective magnetic field:

where S=(Subscript[I, a])ω is the spin angular momentum (magnitude). The magnetic field points towards the origin, where there is effective an magnetic monopole.

Block[{th=0.014,dashedtube},
dashedtube[v_,sp_:0.1]:=Tube[Table[{x v,(x+0.05)v},{x,0,0.9,sp}],th];Graphics3D[{{Opacity[0.5],White,Sphere[{0,0,0},1]},{Blue,Sphere[{0,0,0},0.1],Arrowheads[0.035],Table[Block[{θ,ϕ},θ=ArcCos[1-2x];ϕ=20θ;Arrow[Tube[{Sin[θ]Cos[ϕ],Sin[θ]Sin[ϕ],Cos[θ]}#&/@{1.02,1.25},0.015]]],{x,0,1,0.01}],Text["\!\(-B\)",{0,-1,1}]},Block[{θ=π/4},{{Black,dashedtube[{0,0,1}],dashedtube[{0,Sin[θ],Cos[θ]}],Translate[dashedtube[{0,Sin[θ],0},0.15],{0,0,Cos[θ]}],Tube[0.2{0,Sin[#],Cos[#]}&/@Range[0,θ,π/32],th],Text["\!\(θ\)",{0,0.3,0.1}],Text["\!\(sin θ\)",{0,0.3,0.8}]},{Red,Arrowheads[{0.04,#+0.01}&/@(Range[0,5]/6)],Arrow[Tube[Table[{Sin[θ]Cos[ϕ],Sin[θ]Sin[ϕ],Cos[θ]},{ϕ,0,2π,π/32}],th]]}}]},Boxed->False,Lighting->"Neutral",ViewPoint->{5,0,2},ImageSize->140]]

One common choice of vector potential A(n) to produce such magnetic field is the Wu-Yang monopole potential []

Verify that A(n) given in Eq. () satisfies Eq. ().

(Subscript[A, θ])=0,(Subscript[A, φ])=S ((cos θ-1)/(sin θ)).

(Subscript[∇, ⊥])×A=(1/(sin θ))((∂_θ)((Subscript[A, φ])sin θ)-(∂_φ)(Subscript[A, θ]))n
=(1/(sin θ))(∂_θ)(S ((cos θ-1)/(sin θ))sin θ)n
=(S/(sin θ))(∂_θ)(cos θ-1)n
=((-sin θ)/(sin θ))S n
=-S n.

As a simple justification, along a latitude loop at the polar angle θ, the loop integral of the vector potential is

∮A·dl=∫S ((cos θ-1)/(sin θ))sin θdφ=2π S (cos θ-1),

which indeed equals to the magnetic flux through the spherical cap from the north pole down to angle θ

∫B·dσ=-S Ω(θ)=-2π S (1-cos θ),

where Ω(θ)=2π (1-cos θ) denotes the solid angle of the cap.

However, as θ->π near the south pole, (Subscript[A, φ])→∞ diverges. What is going wrong?

T. T. Wu, C. N. Yang. Dirac monopole without strings: monopole harmonics. Nuclear Physics B107 365-380 (1976).

Quantization of Spin (or Monopole)

Θ=(1/ℏ)∮A·dl=((2π)/ℏ) sin θ (Subscript[A, φ])=((4π B)/ℏ)=-((4π S)/ℏ).

Θ=2π n,

Therefore, it is only possible to avoid singular assignment of A(n) if the spin angular momentum S is quantized to

with n=0,1,2,….

This is also a statement about the magnetic monopole, that the total magnetic flux emitted by a magnetic monopole must be quantized to

On the northern hemisphere (θ∈[0,π/2], excluding the south pole):

(Subscript[A, N])(n)=S ((cos θ-1)/(sin θ))(Subscript[e, φ]).

On the southern hemisphere (θ∈[π/2,π], excluding the north pole):

(Subscript[A, S])(n)=S ((cos θ+1)/(sin θ))(Subscript[e, φ]).

On the equator (θ=π/2) where both patches overlap, the two gauge potentials are related by a gauge transformation:

(Subscript[A, N])(φ)-(Subscript[A, S])(φ)=ℏ (Subscript[e, φ])(∂_φ)χ(φ),

with χ(φ)=-2(S/ℏ) φ.

(e^(i χ(φ)))=(e^(i χ(φ+2π)))
=>exp(-i 2(S/ℏ) φ)=exp(-i 2(S/ℏ) (φ+2π))

((2S)/ℏ)∈ℤ,

Quantum Spin

Hamiltonian

(Subscript[I, t])=m - the transverse moment of inertia of the spin,

(Subscript[D, ⊥]) - surface covariant derivative, along tangent directions on the sphere,

(Subscript[∇, ⊥])=(e^θ)(∂_θ)+(e^φ)(1/(sin θ))(∂_φ) - surface gradient,

A=(Subscript[A, θ])(e^θ)+(Subscript[A, φ])(e^φ) - gauge connection (vector potential) on the sphere. Take the Wu-Yang monopole potential:

(Subscript[A, θ])=0,(Subscript[A, φ])=S ((cos θ-1)/(sin θ)),

with the quantization condition (n∈ℕ),

S=ℏ s=ℏ(n/2),

where s=n/2 is introduced as the spin quantum number (quantized to half integers), also characterizing the monopole strength.

(Subscript[D, ⊥]^2)ψ=(1/(sin θ))((∂_θ)(sin θ (∂_θ))+(1/(sin θ))((((∂_φ)+i s (1-cos θ)))^2))ψ.

Derive Eq. () using differential geometry approach.

A=(Subscript[A, φ])sin θdφ.

(Subscript[D, ⊥])→D=d-(i/ℏ)A=d-((i (Subscript[A, φ]))/ℏ)sin θdφ.

(Subscript[D, ⊥]^2)ψ=★ D ★ D ψ
=★ D ★ (dψ-((i (Subscript[A, φ]))/ℏ)sin θ ψdφ)
=★ D ★ ((∂_θ)ψdθ+((∂_φ)ψ-((i (Subscript[A, φ]))/ℏ)sin θ ψ)dφ)
=★ D (sin θ (∂_θ)ψdφ-(1/(sin θ))((∂_φ)-((i (Subscript[A, φ]))/ℏ)sin θ)ψdθ)
=★ (d-((i (Subscript[A, φ]))/ℏ)sin θdφ)(sin θ (∂_θ)ψdφ-(1/(sin θ))((∂_φ)-((i (Subscript[A, φ]))/ℏ)sin θ)ψdθ)
=★ ((∂_θ)(sin θ (∂_θ)ψ)dθ⋀dφ-(1/(sin θ))((∂_φ)-((i (Subscript[A, φ]))/ℏ)sin θ)((∂_φ)-((i (Subscript[A, φ]))/ℏ)sin θ)ψdφ⋀dθ)
=★ ((∂_θ)(sin θ (∂_θ)ψ)+(1/(sin θ))((((∂_φ)-((i (Subscript[A, φ]))/ℏ)sin θ))^2)ψ)dθ⋀dφ
=(1/(sin θ))((∂_θ)(sin θ (∂_θ)ψ)+(1/(sin θ))((((∂_φ)-((i (Subscript[A, φ]))/ℏ)sin θ))^2)ψ)
=(1/(sin θ))((∂_θ)(sin θ (∂_θ))+(1/(sin θ))((((∂_φ)-((i (Subscript[A, φ]))/ℏ)sin θ))^2))ψ.

(Subscript[D, ⊥]^2)ψ=(1/(sin θ))((∂_θ)(sin θ (∂_θ))+(1/(sin θ))((((∂_φ)-((i (Subscript[A, φ]))/ℏ)sin θ))^2))ψ
=(1/(sin θ))((∂_θ)(sin θ (∂_θ))+(1/(sin θ))((((∂_φ)-((i S)/ℏ)(cos θ-1)))^2))ψ.

(Subscript[D, ⊥]^2)ψ=(1/(sin θ))((∂_θ)(sin θ (∂_θ))+(1/(sin θ))((((∂_φ)+i s (1-cos θ)))^2))ψ.

Schrödinger Equation

Separation of variables: let

ψ(θ,φ)=(e^(i (m-s) φ))ψ(θ),

Eq. () takes the form of the generalized associated Legendre differential equation

((1/(sin θ))(∂_θ)(sin θ (∂_θ))-(1/((sin^2) θ))(((m-s cos θ))^2)+λ)ψ(θ)=0

where the eigenvalue λ is related to the energy E by

E=((ℏ^2)/(2(Subscript[I, t])))λ+((ℏ^2)/(2(Subscript[I, a])))(s^2).

Justify Eq. () and Eq. ().

(OverHat[H])ψ(θ,φ)=((1/(2(Subscript[I, t])))(((-i ℏ (Subscript[D, ⊥])))^2)+((S^2)/(2(Subscript[I, a]))))ψ(θ,φ)
=-((ℏ^2)/(2(Subscript[I, t])))(Subscript[D, ⊥]^2)ψ(θ,φ)+(((ℏ^2)(s^2))/(2(Subscript[I, a])))ψ(θ,φ).

(Subscript[D, ⊥]^2)ψ(θ,φ)=(Subscript[D, ⊥]^2)(e^(i (m-s) φ))ψ(θ)
=((1/(sin θ))(∂_θ)(sin θ (∂_θ))+(1/((sin^2) θ))((((∂_φ)+i s (1-cos θ)))^2))(e^(i (m-s) φ))ψ(θ)
=(e^(i (m-s) φ))((1/(sin θ))(∂_θ)(sin θ (∂_θ))+(1/((sin^2) θ))(((i(m-s)+i s (1-cos θ)))^2))ψ(θ)
=(e^(i (m-s) φ))((1/(sin θ))(∂_θ)(sin θ (∂_θ))-(1/((sin^2) θ))(((m-s+s-s cos θ))^2))ψ(θ)
=(e^(i (m-s) φ))((1/(sin θ))(∂_θ)(sin θ (∂_θ))-(1/((sin^2) θ))(((m-s cos θ))^2))ψ(θ).

((1/(sin θ))(∂_θ)(sin θ (∂_θ))-(1/((sin^2) θ))(((m-s cos θ))^2))ψ(θ)=-λ ψ(θ),

(Subscript[D, ⊥]^2)ψ(θ,φ)=(e^(i (m-s) φ))(-λ)ψ(θ)
=-λ(e^(i (m-s) φ))ψ(θ)
=-λ ψ(θ,φ).

(OverHat[H])ψ(θ,φ)=-((ℏ^2)/(2(Subscript[I, t])))(-λ)ψ(θ,φ)+(((ℏ^2)(s^2))/(2(Subscript[I, a])))ψ(θ,φ)
=(((ℏ^2)/(2(Subscript[I, t])))λ+(((ℏ^2)(s^2))/(2(Subscript[I, a]))))ψ(θ,φ),

E=((ℏ^2)/(2(Subscript[I, t])))λ+(((ℏ^2)(s^2))/(2(Subscript[I, a]))).

The equation Eq. () can be solved by

ψ(θ)=(((1-cos θ))^((s-m)/2))(((1+cos θ))^((s+m)/2))(Subscript[P, l-s]^((s-m,s+m)))(cos θ),

where

(Subscript[P, n]^((a,b)))(x) denotes the Jacobi polynomial,

(Subscript[P, n]^((a,b)))(x)=(2^(-n))(∑_(k=0)^n)(((n+a)!)/(k!(n+a-k)!))(((n+b)!)/((n-k)!(b+k)!))(((x-1))^(n-k))(((x+1))^k).

which is well-defined for x∈[-1,1] if n,n+a,n+b∈ℕ, implying the following quantization conditions:

l=s,s+1,s+2,…,

m=-l,-l+1,…,l-1,l.

The corresponding eigenvalue λ is

λ=l (l+1)-(s^2).

Verify Eq. () and Eq. ().

(∂_θ)=((∂ cos θ)/(∂θ))(∂/(∂ cos θ))=-sin θ (∂_x).

ψ(θ)=F(cos θ)=F(x).

(((-sin θ)/(sin θ))(∂_x)(-(sin^2) θ (∂_x))-(1/((sin^2) θ))(((m-s cos θ))^2)+λ)F(x)
=((∂_x)((1-(x^2))(∂_x))-((((m-s x))^2)/(1-(x^2)))+λ)F(x)=0.

F(x)=(((1-x))^((s-m)/2))(((1+x))^((s+m)/2))P(x),

Block[{F},
F[x_]:=(1-x)^((s-m)/2)(1+x)^((s+m)/2)P[x];
Collect[#,{P[x],P'[x],P''[x]}]&/@Factor[D[(1-x^2)D[F[x],x],x]-(m-s x)^2/(1-x^2)F[x]+λ F[x]]]

((1-(x^2))(∂_x^2)+(2m-2(1+s)x)(∂_x)+(λ-s))P(x)=0.

((1-(x^2))(∂_x^2)+(b-a-(a+b+2)x)(∂_x)+n(n+a+b+1))P(x)=0,

2m=b-a,
2(1+s)=a+b+2,
λ-s=n(n+a+b+1)

Solve[{b-a==2m,b+a+2==2(1+s),λ-s==n(n+a+b+1)},{a,b,λ}]

a=s-m,
b=s+m,
λ=(n^2)+(2s+1)n+s.

Simplify[n+(n^2)+s+2 n s/.n->l-s]

λ=l(l+1)-(s^2).

Monopole Harmonics

The eigen wavefunction of (OverHat[H]) is given by

The corresponding eigen energy is

In the isotropic limit ((Subscript[I, a])=(Subscript[I, t])=I), the eigen energy only depends on the quantum number l,

(Subscript[E, s,l,m])=((ℏ^2)/(2I))l (l+1).

The quantum numbers s,l,m take values in

s=0,1/2,1,3/2,…∈ℕ/2, with S=ℏ s => sets the monopole strength (i.e. total magnetic flux through the unit sphere).

l=s,s+1,s+2,…∈s+ℕ => labels the Landau levels on the sphere.

m=-l,-l+1,…,l-1,l => labels the degenerated states within each Landau level => Landau level-l has the degeneracy (2l+1).

Angular Momentum

S=S n=ℏ s n is the spin angular momentum (axial component).

n×(-i ℏ (Subscript[D, ⊥])) is the orbital angular momentum (transverse component) associated with the spinning axis precession, which is given by the cross product between:

n - axis coordinate (on the sphere),

-i ℏ (Subscript[D, ⊥]) - axis kinetic momentum (in the tangent plane).

(OverHat[L])=ℏ ((e^φ)(-i(∂_θ))-(e^θ)(1/(sin θ))((-i(∂_φ))+s (1-cos θ))+s n).

Derive Eq. ().

(OverHat[L])=n×(-i ℏ (Subscript[D, ⊥]))+S
=-i ℏ n×((Subscript[∇, ⊥])-(i/ℏ)A)+ℏ s n
=-i ℏ n×((e^θ)(∂_θ)+(e^φ)(1/(sin θ))(∂_φ)-(i/ℏ)S ((cos θ-1)/(sin θ))(e^φ))+ℏ s n
=-i ℏ n×((e^θ)(∂_θ)+(e^φ)(1/(sin θ))((∂_φ)+i s (1-cos θ)))+ℏ s n
=-i ℏ((e^φ)(∂_θ)-(e^θ)(1/(sin θ))((∂_φ)+i s (1-cos θ)))+ℏ s n
=ℏ ((e^φ)(-i(∂_θ))-(e^θ)(1/(sin θ))((-i(∂_φ))+s (1-cos θ))+s n)

n=cos φ sin θ (e^x)+sin φ sin θ (e^y)+cos θ (e^z),
(e^θ)=cos φ cos θ (e^x)+sin φ cos θ (e^y)-sin θ (e^z),
(e^φ)=-sin φ (e^x)+cos φ (e^y).

(Subscript[(OverHat[L]), x])=ℏ (sin φ (i(∂_θ))+cos φ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ)))),
(Subscript[(OverHat[L]), y])=ℏ (-cos φ (i(∂_θ))+sin φ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ)))),
(Subscript[(OverHat[L]), z])=ℏ ((-i(∂_φ))+s ).

Derive Eq. ().

(Subscript[(OverHat[L]), x])=(e^x)·(OverHat[L])
=ℏ ((e^x)·(e^φ)(-i(∂_θ))-(e^x)·(e^θ)(1/(sin θ))((-i(∂_φ))+s (1-cos θ))+s (e^x)·n)
=ℏ (-sin φ (-i(∂_θ))-cos φ ((cos θ)/(sin θ))((-i(∂_φ))+s (1-cos θ))+s cos φ sin θ)
=ℏ (sin φ (i(∂_θ))+cos φ cot θ (i(∂_φ))+s cos φ ((1-cos θ)/(sin θ)))
=ℏ (sin φ (i(∂_θ))+cos φ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ)))).

(Subscript[(OverHat[L]), y])=(e^y)·(OverHat[L])
=ℏ ((e^y)·(e^φ)(-i(∂_θ))-(e^y)·(e^θ)(1/(sin θ))((-i(∂_φ))+s (1-cos θ))+s (e^y)·n)
=ℏ (cos φ (-i(∂_θ))-sin φ ((cos θ)/(sin θ))((-i(∂_φ))+s (1-cos θ))+s sin φ sin θ)
=ℏ (-cos φ (i(∂_θ))+sin φ cot θ (i(∂_φ))+s sin φ ((1-cos θ)/(sin θ)))
=ℏ (-cos φ (i(∂_θ))+sin φ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ)))).

(Subscript[(OverHat[L]), z])=(e^z)·(OverHat[L])
=ℏ ((e^z)·(e^φ)(-i(∂_θ))-(e^z)·(e^θ)(1/(sin θ))((-i(∂_φ))+s (1-cos θ))+s (e^z)·n)
=ℏ (-(-sin θ)(1/(sin θ))((-i(∂_φ))+s (1-cos θ))+s cos θ)
=ℏ ((-i(∂_φ))+s ).

Verify Eq. ().

[(∂_x),f(x)]=(∂_x)f(x),

[(∂_x),f(x)]ψ(x)=(∂_x)(f(x)ψ(x))-f(x)(∂_x)ψ(x)
=(∂_x)f(x)ψ(x)+f(x)(∂_x)ψ(x)-f(x)(∂_x)ψ(x)
=(∂_x)f(x)ψ(x),

[(Subscript[(OverHat[L]), x]),(Subscript[(OverHat[L]), y])]=(ℏ^2)[sin φ (i(∂_θ))+cos φ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ))),-cos φ (i(∂_θ))+sin φ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ)))]
=(ℏ^2)([sin φ (i(∂_θ)),-cos φ (i(∂_θ))]+[sin φ (i(∂_θ)),sin φ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ)))]+[cos φ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ))),-cos φ (i(∂_θ))]+[cos φ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ))),sin φ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ)))])
=(ℏ^2)(0+(sin^2) φ [i(∂_θ),cot θ (i(∂_φ))+s((1-cos θ)/(sin θ))]+sin φ cot θ [sin φ,i(∂_φ)](i(∂_θ))+cos φ cot θ [i(∂_φ),-cos φ ](i(∂_θ))-(cos^2) φ [cot θ (i(∂_φ))+s((1-cos θ)/(sin θ)),i(∂_θ)]+cos φ cot θ[ i(∂_φ),sin φ ](cot θ (i(∂_φ))+s((1-cos θ)/(sin θ)))+sin φ cot θ [cos φ,i(∂_φ)](cot θ (i(∂_φ))+s((1-cos θ)/(sin θ))))
=(ℏ^2)(i (sin^2) φ ((1/((sin^2)θ))(-i(∂_φ))+(s/(1+cos θ)))-i sin φ cos φ cot θ (i(∂_θ))+i sin φ cos φ cot θ (i(∂_θ))+i (cos^2) φ ((1/((sin^2)θ))(-i(∂_φ))+(s/(1+cos θ)))+i (cos^2) φ cot θ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ)))+i (sin^2) φ cot θ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ))))
=i (ℏ^2)(((1/((sin^2)θ)) (-i(∂_φ))+(s/(1+cos θ)))+cot θ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ))))
=i (ℏ^2)(((1/((sin^2)θ))-(cot^2) θ)(-i(∂_φ))+s ((1/(1+cos θ))+(((1-cos θ)cos θ)/((sin^2) θ))))
=i (ℏ^2)((-i(∂_φ))+s)
=i ℏ (Subscript[(OverHat[L]), z]).

[(Subscript[(OverHat[L]), z]),(Subscript[(OverHat[L]), x])]=(ℏ^2)[(-i(∂_φ))+s ,sin φ (i(∂_θ))+cos φ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ)))]
=(ℏ^2)(-i [(∂_φ) ,sin φ](i(∂_θ))-i [(∂_φ),cos φ](cot θ (i(∂_φ))+s((1-cos θ)/(sin θ))))
=i (ℏ^2)(-cos φ (i(∂_θ))+sin φ(cot θ (i(∂_φ))+s((1-cos θ)/(sin θ))))
=i ℏ (Subscript[(OverHat[L]), y]).

sin φ→sin(φ-π/2)=-cos φ,
cos φ→cos(φ-π/2)=sin φ,
(Subscript[(OverHat[L]), x])→(Subscript[(OverHat[L]), y]),

[(Subscript[(OverHat[L]), z]),(Subscript[(OverHat[L]), y])]=i (ℏ^2)(-sin φ (i(∂_θ))-cos φ (cot θ (i(∂_φ))+s((1-cos θ)/(sin θ))))
=-i ℏ (Subscript[(OverHat[L]), x]),

Squared Angular Momentum

For our case of (OverHat[L]) in Eq. (), ((OverHat[L])^2) can be explicitly written out

((OverHat[L])^2)=-(ℏ^2)(Subscript[D, ⊥]^2)+(S^2)
=((-(ℏ^2))/(sin θ))((∂_θ)(sin θ (∂_θ))+(1/(sin θ))((((∂_φ)+i s (1-cos θ)))^2))+(ℏ^2)(s^2).

A key property of ((OverHat[L])^2) is that it commutes with any component of the angular momentum operator,

or simply written as [((OverHat[L])^2),(OverHat[L])]=0 in vector form. It is commonly referred to as the Casimir operator of the 𝔰𝔬(3) algebra — an element in the Lie algebra that commutes with all its generators.

Proof Eq. () based on Eq. ().

[((OverHat[L])^2),(Subscript[(OverHat[L]), a])]=[(Subscript[(OverHat[L]), b])(Subscript[(OverHat[L]), b]),(Subscript[(OverHat[L]), a])]
=(Subscript[(OverHat[L]), b])[(Subscript[(OverHat[L]), b]),(Subscript[(OverHat[L]), a])]+[(Subscript[(OverHat[L]), b]),(Subscript[(OverHat[L]), a])](Subscript[(OverHat[L]), b])
=i ℏ (ϵ^(b,a,c))(Subscript[(OverHat[L]), b])(Subscript[(OverHat[L]), c])+i ℏ (ϵ^(b,a,c))(Subscript[(OverHat[L]), c])(Subscript[(OverHat[L]), b])
=i ℏ ((ϵ^(b,a,c))+(ϵ^(c,a,b)))(Subscript[(OverHat[L]), b])(Subscript[(OverHat[L]), c])
=i ℏ (-(ϵ^(a,b,c))+(ϵ^(a,b,c)))(Subscript[(OverHat[L]), b])(Subscript[(OverHat[L]), c])
=0.

Verify Eq. ().

(Subscript[D, ⊥]^2)(Subscript[Y, s,l,m])(θ,φ)=-λ(Subscript[Y, s,l,m])(θ,φ)=-(l (l+1)-(s^2))(Subscript[Y, s,l,m])(θ,φ),

((OverHat[L])^2)(Subscript[Y, s,l,m])(θ,φ)=(-(ℏ^2)(Subscript[D, ⊥]^2)+(S^2))(Subscript[Y, s,l,m])(θ,φ)
=((ℏ^2)(l (l+1)-(s^2))+(ℏ^2)(s^2))(Subscript[Y, s,l,m])(θ,φ)
=(ℏ^2)l (l+1)(Subscript[Y, s,l,m])(θ,φ).

(Subscript[Y, s,l,m])(θ,φ)=(e^(i (m-s) φ))(Subscript[Θ, s,l,m])(θ),

(Subscript[(OverHat[L]), z])(Subscript[Y, s,l,m])(θ,φ)=ℏ ((-i(∂_φ))+s )(e^(i (m-s) φ))(Subscript[Θ, s,l,m])(θ)
=ℏ ((m-s)+s )(e^(i (m-s) φ))(Subscript[Θ, s,l,m])(θ)
=ℏ m (Subscript[Y, s,l,m])(θ,φ).

The quantum numbers are reinterpreted as

l=s,s+1,s+2,…∈s+ℕ - angular quantum number (labeling quantized total angular momentum),

m=-l,-l+1,…,l-1,l - magnetic quantum number (labeling quantized z-component of angular momentum).

Spin-1/2

s=1/2,

S=ℏ s=(ℏ/2).

There are only two options for the quantum number m

m=±(1/2),

corresponding to the up and down spin states.

The corresponding monopole harmonics wave functions are

(Subscript[Y, 1/2,1/2,m])(θ,φ)=(1/(Sqrt[4π]))(((e^(i (m-1/2) φ))(1-cos θ))^((1/2-m)/2))(((1+cos θ))^((1/2+m)/2)),

or respectively as

Block[{monoY},
monoY[s_,l_,m_]:=SphericalPlot3D[1,{θ,0,π},{φ,0,2π},ColorFunction->Function[{x,y,z,θ,φ,r},ComplexColor[Exp[I(m-s)φ]((1-Cos[θ])/2)^((s-m)/2)((1+Cos[θ])/2)^((s+m)/2)JacobiP[l-s,s-m,s+m,Cos[θ]]]],Axes->False,SphericalRegion->True,Boxed->False,ColorFunctionScaling->False,Lighting->"Neutral",ViewPoint->{-5,2,0},ImageSize->140];
Grid[{Style[#,"InlineFormula"]&/@{"\!\(Y\_\(1/2,1/2,+\(1/2\)\)\)","\!\(Y\_\(1/2,1/2,-\(1/2\)\)\)"},{monoY[1/2,1/2,1/2],monoY[1/2,1/2,-1/2]}}]]

Angular momentum eigenvalues:

((OverHat[L])^2)(Subscript[Y, 1/2,1/2,±1/2])=(3/4)(ℏ^2)(Subscript[Y, 1/2,1/2,±1/2]),
(Subscript[(OverHat[L]), z])(Subscript[Y, 1/2,1/2,±1/2])=±(1/2)ℏ (Subscript[Y, 1/2,1/2,±1/2]).

Energy eigenvalue (2-fold degenerated)

(OverHat[H])(Subscript[Y, 1/2,1/2,±1/2])=(((ℏ^2)/(4(Subscript[I, t])))+((ℏ^2)/(8(Subscript[I, a]))))(Subscript[Y, 1/2,1/2,±1/2]).

Topological obstruction: Monopole harmonics can not be globally single-valued in a naive coordinate sense because of the nontrivial gauge curvature induced by the monopole.

However, the apparent multivaluedness is a gauge artifact, and can be removed by the gauge transformation.

For the northern hemisphere (θ∈[0,π/2], excluding the south pole):

(Subscript[A, N])(θ,φ)=(ℏ/2)((cos θ-1)/(sin θ))(Subscript[e, φ]),
(Subscript[Y, 1/2,1/2,+1/2]^N)(θ,φ)=(1/(Sqrt[4π]))(Sqrt[1+cos θ]),
(Subscript[Y, 1/2,1/2,-1/2]^N)(θ,φ)=(1/(Sqrt[4π]))(e^(-i φ))(Sqrt[1-cos θ]).

For the southern hemisphere (θ∈[π/2,π], excluding the north pole):

(Subscript[A, S])(θ,φ)=(ℏ/2)((cos θ+1)/(sin θ))(Subscript[e, φ]),
(Subscript[Y, 1/2,1/2,+1/2]^S)(θ,φ)=(1/(Sqrt[4π]))(e^(i φ))(Sqrt[1+cos θ]),
(Subscript[Y, 1/2,1/2,-1/2]^S)(θ,φ)=(1/(Sqrt[4π]))(Sqrt[1-cos θ]).

On the equator (θ=π/2), the two gauge choices are related by a gauge transformation:

(Subscript[A, N])(π/2,φ)=(Subscript[A, S])(π/2,φ)+ℏ (Subscript[e, φ])(∂_φ)χ(φ),
(Subscript[Y, 1/2,1/2,m]^N)(π/2,φ)=(e^(i χ(φ)))(Subscript[Y, 1/2,1/2,m]^S)(π/2,φ),

with χ(φ)=-φ.

Verify Eq. ().

Check

(Subscript[A, N])(π/2,φ)=(ℏ/2)((cos(π/2)-1)/(sin(π/2)))(Subscript[e, φ])
=-(ℏ/2)(Subscript[e, φ]),

(Subscript[A, S])(π/2,φ)+ℏ (Subscript[e, φ])(∂_φ)χ(φ)=(ℏ/2)((cos(π/2)+1)/(sin(π/2)))(Subscript[e, φ])+ℏ (Subscript[e, φ])(∂_φ)(-φ)
=(ℏ/2)(Subscript[e, φ])-ℏ (Subscript[e, φ])
=-(ℏ/2)(Subscript[e, φ]),

thus (Subscript[A, N])(π/2,φ)=(Subscript[A, S])(π/2,φ)+ℏ (Subscript[e, φ])(∂_φ)χ(φ).

It can be shown that

(e^(i χ(φ)))(Subscript[Y, 1/2,1/2,m]^S)(π/2,φ)
=(e^(-i φ))(1/(Sqrt[4π]))(((e^(i (m+1/2) φ))(1-cos(π/2)))^((1/2-m)/2))(((1+cos(π/2)))^((1/2+m)/2))
=(1/(Sqrt[4π]))(e^(i (m-1/2) φ))
=(1/(Sqrt[4π]))(((e^(i (m-1/2) φ))(1-cos(π/2)))^((1/2-m)/2))(((1+cos(π/2)))^((1/2+m)/2))
=(Subscript[Y, 1/2,1/2,m]^N)(π/2,φ).

(Subscript[Y, 1/2,1/2,m])(θ,φ)=({{{, ({{(Subscript[Y, 1/2,1/2,m]^N)(θ,φ), θ∈[0,π/2],}, {(Subscript[Y, 1/2,1/2,m]^S)(θ,φ), θ∈[π/2,π],}})}})