Top page (correct Bohr model including helium. )

Strange "spin" is NOT a real thing

Spin Hall effect doesn't mean "spin".

Calculation of various values.

- Topological insulator is math "trick".
- Reason why Monopole is impossible.
- Integer and fractional Hall effect.
- Massless Dirac fermion really exists ?
- Half integer Hall effect of graphene.

*(Fig.1) Topological insulator ?*

Topological insulators are very special composite materials such as Hg(Cd)Te, behaves as an insulator in its interior but whose surface ( or **edge** in 2D system ) contains conducting states. ( See Wiki )

They insist, in 2-dimensional topological insulator, "up" and "down" spin electrons are moving in opposite directions along the **edge**. ( So each spin cannot be detected )

But of course, unrealistic spin cannot be observed directly.

As I say later, direct experimental result of this topological insulator is based only on **conductivity** ( NOT spin ).

*(Fig.2) "Time reversal symmetry" is the main reason for topological insulator ? ← Physics ?*

Then what on earth causes this topological insulator ?

According to the current standard quantum mechanics, the main **mechanism** is supposed to be "Time reversal **symmetry**".

In Fig.2, "up" spin electron is moving rightward.

If you **reverse** time ( forwad and backward ) directions, spinning direction becomes opposite ( clockwise → counterclockwise ), which means "up" and "down" spins are reversed.

Furthermore, moving direction of electron also becomes opposite ( rightward → leftward ).

As I said in Fig.1, topological insulator contains both these spin states, they insist.

So time reversal symmetry **doesn't** break this bidirectional state.

But as you notice, they just say this state as it is, and do **NOT** say about the **detailed** mechanism itself.

The current quantum mechanics is **only** satisfied with very vague math symbols, which **cannot** be called "physics".

*(Fig.3) "Massless" Dirac fermion is real ? or only illusion ?*

Surprisingly, the current physicists argue that **massless** Dirac fermions **exist** in topological insulator and graphene, as shown on this site. Of course, "massless" Dirac fermions are completely **unreal** concepts.

They measure the relation of momentum and energy of **electrons** ejected by some lasers using ARPES.

But as you know, each electron actually **has** "mass".

So their conclusion of **massless** Dirac fermions based on **massive** electrons are completely **contradictory**.

Surprisingly, even top journals believe this **unrealistic** massless fermions, as shown on this site.

As I explain later, the experiments using ARPES is based on **strange** assumption, and their results are very **vague**. ( See Dirac cone ).

*(Fig.4) Magnetic monopole is real ?*

Magnetic **monopole** is hypothetical particle that is an isolated magnet with only one magnetic pole ( a north pole without a south pole or vise-versa ), as shown in Wiki.

Of course, this magnetic monopole is only **fiction**, and there is **NO** experimental evidence at all.

Because the magnetic field is caused by the movement of **electric** charge ( NOT by **fictitious** magnetic charge ).

*(Fig.5) Magnetic monopole is real ? Only "mathematical" trick ?*

Though, there is **NO** experimental evidence of monopole, the current physicists try to apply this monopole in actual quantum Hall effect, as shown on this site. this site (p.3 right), and **10**-dimensional string theory.

Because the current quantum mechanics tends to avoid real physical world, and escape into fictitious math tricks, as shown in quasiparticle.

As I explain later, we can prove "monopole" itself is **impossible** from both physical and mathematical viewpoints.

If we are bound with mathematical **tricks**, the real science would **NEVER** advance, forever.

*(Fig.6) Berry phase is virtual ( NOT real ! ) phase. ← Physics ?*

Berry phase ( see this site and this site ) is **virtual** phase, which was introduced for explaining various physical phenomena such as ( anomalous ) quantum Hall effect.

Even if you ask some specialists, "Berry phase really exists ?", they **cannot** give clear answer at all.

As I say later, this Berry phase is based on **wrong** and artificial assumtions in both physical and mathematical aspects. So there are **NO** realities here.

This means the mechanism of quantum Hall effect remains **unknown** ( realisitically ) even now.

*(Fig.7) Several vortices of magnetic flux (= Φ ) are "attached" to each electron ?*

In quantum Hall effect, each electron moves in the direction **perpendicular** to the external electric field.

So, magnetic Lorentz force (= ev× B ) is equal to electric field (= eE ) in the equilibrium state.

In the fractional quantum Hall effect (= FQHE ), Hall conductance becomes fractional values of e^{2}/h, as shown in Wiki.

Usually, magnetic flux (= Φ ) is expressed as magnetic flux density (= B ) × area (= S ), so Φ = BS.

But in fractional QHE, they argue the **point**-like electron contains several magnetic flux **inside** it, as shown in Fig.7 right, and this site, this site, this site (Fig.6).

From the realistic viewpoint, it is **impossible** that several magnetic flux exist inside the **point**-like area.

*(Fig.8) Anyon = fractional charge really exists ? *

Surprisingly, as shown on this site, this site, this site, the researchers argue **fractional** charges such as 1/3e, 1/2e .. really exist.

Of course, these fractional charges are completely **unreal** concepts.

These are Only one of **fictitious** quasiparticles.

The problem is that the current "Shut up and calculate !" physics **doesn't** try to ask what virtual quasiparticles really are, "forever".

*(Fig.9) Magnetic monolole doesn't really exist.*

Magnetic monopole is hypothetical particle, which does **NOT** really exist.

Magnetic charge consists Only of north ( or south ) pole, like electric charge.

The problem is the "**fantasy**" quantum mechanics and cosmology try to use this **illusory** monopole in (spin) quantum Hall effect, journal, and early universe.

*(Fig.10) Magnetic flux density (= B ), magnetic flux (= Φ ), vector potential (= A ). *

Magnetic flux ( field ) density per area can be expressed using vector ( magnetic ) potential A, as shown on this site and this site.

*(Eq.1) B = curl of A.*

Magnetic flux (= Φ ) is given by the **surface** (= S ) integral of the B-field through the surface.

As a result, the total magnetic flux is equal to the **line**-integral of the vector potential "A" along a closed contour (= C ) bounding the surface ( see this site ).

*(Eq.2) *

*(Fig.11) Magnetic dipole and monopole. Only "dipole" is real. *

If magnetic monopole really exists, the magnetic field B can be expressed using the **magnetic charge** (= q_{m} ), shown in Eq.3.

*(Eq.3) Magnetic monopole q _{m}*

In Eq.3, the magnetic charge × the distance (= l ) between ± q_{m} is magnetic **dipole** moment (= m ).

This magnetic dipole moment is equal to the electric current I × area (= S ), which really exists.

*(Eq.4) Vector potential "A" around monopole.*

As shown on this site (p.9) and this site, the vector potential A around this monopole becomes like Eq.4.

*(Fig.12) Total magnetic flux included in one part (= S ) of the spherical surface. *

Total magnetic flux ( Φ ) included in the upper surface ( 0 < angle < θ ) becomes like Fig.12.

The line integral of A (= Eq.4) and the surface integral of B (= Eq.3 ) in this area becomes just the same, as follows,

*(Eq.5) The relation between vector potential "A" and magnetic field "B".*

So you understand the form of Eq.4 is right.

*(Fig.13) Magnetic field B at θ = π diverges to infinity ( ∞ ) ! *

But there is one **serious** problem in this form.

As you see in Fig.13, it is known that the vector potential A ( and magnetic field B ) **always** diverges to **infinity** at one point ( in this case, θ = π ).

So Eq.4 is **inconsistent** with magnetic field (= Eq.3 ) caused by monopole.

This is the most important reason why **illusory** magnetic monopole **cannot** really exist even mathematically.

*(Fig.14) Total magnetic flux (= Φ ) included in the upper hemispherical surface. *

As shown in Fig.14, using Eq.2, total magnetic flux through the surface of the upper hemisphere becomes 2πr A.

*(Fig.15) Total magnetic flux (= Φ ) included in the lower hemispherical surface. *

On the other hand, from the viewpoint of lower hemisphere, the sign of vector potential A becomes just **opposite** on the equator line, as shown in Fig.15.

So the total magnetic flux included in the lower hemisphere becomes opposite to Fig.14.

*(Eq.6) Magnetic monopole always becomes zero !?*

The magnetic monopole ( charge, q_{m} ) means the total magnetic flux through **all** sperical surface.

But the sum of Fig.14 (= upper sphere ) and Fig.15 (= lower sphere ) is always cancelled out to **zero**, like Eq.6.

This is the important property of the curl of A.

( Due to the curl, only magnetic dipole exists, monopole **doesn't**. )

*(Fig.16) Monopole (= magnetic charge ) is always ZERO ! *

So, as long as magnetic field is the curl of vector potential ( B = ∇ × A ), magnetic charge is always **cancelled** out, and becomes **zero**.

Why does it seem to be contradictory ?

In fact, because we neglect the **infinity** of Fig.13, this **inconsistency** happens.

Of course, a **singularity** means this monopole is wrong.

*(Fig.17) When the moving direction changes on the equator, A ^{N} → A^{S} !?*

To avoid inconsistency in Fig.16, we have to depend on very **atificial** rules like Fig.17.

In Fig.17, vector potential "A" **suddenly** changes ( A^{N} → A^{S} ) one the borderline between upper and lower hemisphere.

This change is **discontinuous**, so both physically and mathematically, this monopole is **unreasonable** ( see Fig.18 lower ).

*(Eq.7) Magnetic monopole ?*

Using the very **artificial** rule of Fig.17, the non-zero magnetic charge (= q_{m} ) can be gotten as shown in Eq.7. ( See also this site and this site (p.9). )

*(Fig.18) Monopole is very unrealistic. *

So they start to argue the same mechanism of this magnetic monopole can be applied to other actual phenomena.

One of the important phenomena is quantum Hall effect (= QHE ).

*(Fig.19) Quantum ( spin ) Hall effect = monopole !? *

But as you feel, this idea **cannot** be acceptable from the realistic viewpoint.

So the current interpretation about quantum Hall effect is **NOT** correct.

*(Fig.20) What is Berry phase ? = Application of "unreal" Monopole ? *

Berry phase is very **abstract** concept, which often appears in quantum Hall effect.

As shown in Fig.20, they **artificially** add other mathematical **phase** (= γ(t) ) to the original eigenfunction.

In the original eigenfunction, the exponential function representing **time** evolution means **Hamiltonian** (= energy E ) .

But if it includes Berry phase, this simple relation is NOT satisfied, they insist.

This phase is clearly **inconsistent** with the time evolution rule of basic quantum field theory.

*(Eq.8) Berry phase ?*

Eigenfucntion φ is supposed to depend on some parameter "R" ( this R can be **arbitrary** concept ).

During the time from "0" to "t", this system goes back to the original state.

(= cyclic adiabatic evolution. )

*(Eq.9) Magnetic monopole ?*

The line integral along the arbitrary parameter R is similar to the relation between vector potential A and magnetic field B ( B = ∇ × A ).

So they define new **artificial** vector potential in Eq.9.

This vector-potential like "A" in Berry phase is called "**Berry connection**" or "gauge field".

*(Eq.9') Berry curvature.*

Furthermore, the curl of this "A" is magnetic field (= B ) -like, which is called "Berry **curvature**".

This leads to artificial monopole, as I explain later.

About the detailed calculation of Eq.8, see this page.

*(Fig.21) Two-dimensional electron system.*

Quantum Hall effect is observed in two-dimensional electron systems subjected to low temperatures and strong magnatic fields, in which Hall conductivity σ takes the **quantized** values (= *n e ^{2}/h* ).

See Wiki, this site and this site.

In this system, moving electrons are confined to only one plane (= red part of Fig.21 ).

To achieve this condition, we take the interface between a semiconductor and other material. See also this site.

*(Fig.22) Two-dimensional electron system.*

Under the strong magnetic field (= B ), each electron is circularly (= cyclotron ) moving.

Centrifugal force is equal to magnetic Lorentz force (= ev× B ).

This energy levels are quantized (= **Landau** quantization ).

*(Fig.23) *

As a result, the angular frequency (= ω ) of this circular movement becomes

*(Eq.10)*

If we apply external electric field (= E_{y} ) in the y direction, this electric force (= -eE_{y} ) is equal to the magnetic force (= -ev_{x} × B ).

So each circular orbit moves in the **x** direction.

This Hall conductivity (= σ_{xy} ) is quantized.

*(Eq.11)*

As shown on this page, Hamiltonian changes to Eq.11 under external magnetic field.

"A" is vector ( magnetic ) potential.

*(Eq.12)*

We suppose each component of vector potential A is Eq.12.

In this condition, magnetic field B is in the **z** direction.

Substituting "A" of Eq.12 into Eq.11,

*(Eq.13)*

In Hamiltonian Eq.13, y and z components are free electron.

So we can define wavefunction as

*(Eq.14)*

Considering the momentum is -iħ∇ and using Eq.14, Hamltonian (= energy ) becomes Eq.15 ( in x-y plane ).

*(Eq.15)*

If we solve the Hamiltinian of harmonic oscillator of Eq.16, we can get the quantized energy, as shown on this site.

( Of course, this quantization is related to de Broglie wavelength. )

*(Eq.16)*

where

*(Eq.17)*

Comparing Eq.15 with Eq.16, the angular frequency ω becomes

*(Eq.18)*

Eq.18 is just equal to the result of classical mechanics (= Eq.10 ).

*(Eq.19) "Wave" is quantized in y direction.*

The important point is that electron's **density** (= N_{s} ) is **quantized**, which leads to quantized Hall conductance ( or resistance ).

We suppose the area of this 2D system is L_{x} × L_{y}.

If an integer times **de Broglie** waves are included in each direction, this wavelength λ becomes L_{y}/n in the y direction ( "n" is some integer ).

*(Eq.20)*

The wavenumber "k" is 2π/λ.

As you see, the current quantum mechanics clearly **depends** on quantized de Broglie wave, they **don't** try to admit this fact. This is strange.

From Eq.19, we get Eq.20.

*(Eq.21)*

From Eq.15, the central point of this harmonic oscillator is ħk/eB.

Considering one side length in the x direction is L_{x}, we get Eq.21.

*(Eq.22)*

Substituting the wavenumber k ( in the y direction ) of Eq.20 into Eq.21, we get Eq.22.

So they insist density of state (= N_{s} ) becomes

*(Eq.23)*

Here we divide it by the area L_{x}L_{y}.

Eq.23 can be expressed as

*(Eq.24)*

The result of Eq.24 is equal to this site p.9 and this site p.18-p.20

As you see, they depend on wavenumver "k" space, and this density of state is very **abstract** old concept.

We **cannot** apply this method to more complicated real system.

*(Eq.25)*

Furthermore, they define electron's density (= N_{s} ) in "n" Landau level as Eq.25 .

Of course, this is only **artificial** definition.

From Fig.22, we find the velocity in the y direction becomes Eq.26.

Using Eq.25 and Eq.26, the electric current density (= j_{y} ) becomes

*(Eq.27)*

where ħ = h/2π.

As a result, Hall conductance (= σ_{xy} ) becomes Eq.28.

( Hall conductance means the ratio of electric current in y direction to electric field in x direction. )

*(Eq.28)*

This is the **integral** quantized Hall conductance.

( Hall resistance ρ is a reciprocal of Hall conductance. ρ = 1/σ )

*(Fig.24) Several magnetic flux (= Φ ) are attached to electron ?*

Integer quantum Hall effect (= IQHE ) was first discovered, and they apply the integer (= n ) of Landau levels ( Eq.16 ) under magnetic field to this integer Hall effect.

But **later**, fractional quantum Hall effect (= FQHE ) was discovered.

This is very serious problem.

Because the concept of Landau levels were already used in IQHE, which **cannot** be modified.

So there are **NO** tools left to describe FQHE.

In this strict condition, all they could do was adopting very **strange** idea that several units of magnetic flux (= Φ ) were **attached** to each **point**-like electron, as shown in Fig.24 right and this site, this site, this site (Fig.6).

*(Fig.25) Anyon = fractional charge really exists ? *

This means quasiparticle having fractional charge exist in fractional quantum Hall effect, as shown on this site, this site, this site.

As shown in Eq.25, we can represent electron's density in 2D system as

*(Eq.29) Quantized electron's density.*

When "n" is integer, this becomes integer quantum Hall effect.

In Eq.29, "h/e" is magnetic flux quantum.

Caution: In QHE, they adopt strange flux quantum (= φ_{0} ), which is **twice** the basic one.

This is only an **artificial** rule.

*(Eq.30) Fractional QHE.*

If Eq.29 is true, integer "n" can be a fraction (= 1/l ) in FQHE.

This means each electron's charge (= e ) becomes **fractional** charge (= 1/l e ), as shown in Eq.31.

*(Eq.31)*

Of course, this fractional charge is completely **fantasy**.

These are Only one of **fictitious** quasiparticles.

The most serious problem is that the current "Shut up and calculate !" physics **doesn't** try to ask what virtual quasiparticles really are.

About the theoretical calculation, see also this page.

*(Fig.26) A integer times de Broglie wavelength.*

As you notice, it is very **unnatural** that we adopt completely **different** ( from integer Hall effect ) concepts such as fractional charges and multiple attached flux **only** in fractional Hall effect.

In this section, we prove that these integer and factional Hall effect can be explained naturally by the **common** de Broglie waves.

In Fig.26, each electron is circularly moving under external magnetic field B.

Considering boundary condition (= ends of de Broglie wave fit each other ), the orbital length (= 2π r ) becomes an integer times de Broglie wavelength.

*(Eq.32)*

The first equation means that the centrifugal force is equal to Lorentz force.

The second one is n × de Broglie wavelength ( λ = h/mv ).

In fact, also in quantum mechanics, they depend on an integer times de Broglie wavelength as boundary condition, as shown on this site.

Solving Eq.32, we get

*(Eq.33)*

where, ħ = h/2π

From Eq.33, the area of the circle is

*(Eq.34)*

From Eq.32, the angular frequency ω becomes

*(Eq.35)*

This ω is equal to that of Landau level (= Eq.16, Eq.18 ) of quantum mechanics.

Here we suppose the **k** electrons are included inside the orbit with the area of Eq.34.

*(Fig.27) Two-dimensional electron system.*

As shown in Fig.27, the velocity v_{x} becomes

*(Eq.36)*

Using Eq.34 and Eq.36, the electric current density (J_{x}) in the x direction by the electric field E_{y} is

*(Eq.37)*

From Eq.37, the Hall conductivity σ_{xy} is

*(Eq.38)*

Eq.38 can represent **both** integer and **fractional** quantum Hall effect !

Due to the factor "**2**" in the numerator of Eq.38, the denominator of the irreducible fraction tends to be "*odd*" number.

Hall resistance ρ_{xy} is given by a reciprocal of Hall conductance, so

*(Eq.39)*

Basically, when the diagonal resistance (= ρ_{xx} ) is **zero**, Hall resistance becomes plateau, and the Hall conductivity σ_{xy} (or Hall resistance ρ_{xy} ) is **quantized** including "fractional", as shown in Fig.28.

*(Fig.28) Experimental result.*

What is a concrete state of each 2k/n of Eq.37 and Eq.38 ?

"**n**" means the **number** of de Broglie wavelength in one orbit, and **k** means the number of the **electrons** included inside the orbital circle.

First, we think about **2k/n = 1**. (See also Fig.29.)

(This case corresponds to **n=1** of Eq.28. So "*integer*" quantum Hall effect.)

For example, one orbit of 2 × de Broglie wavelength (**n=2**) contains one electron (**k=1**) like Fig.29.

*(Fig.29) One example of 2k/n = 1. ← integer*

Fig.29 corresponds to n = 1 plateau of Fig.28.

In this case, each orbit is 2 × de Broglie wavelength, so their wave phases tend to attach to each other by **constructive** interference because of the same phase.

Of course, even in 2D system, this layer has some finite **thickness**.

So, several electron's layers may be involved in this electric current.

These cases correspond to n = 1, 2, 3, ... × quantum Hall effect.

*(Fig.30) One example of 2k/n = 2/3. ← fractional.*

Next we consider **2k/n = 2/3**.

This case corresponds to **n= 2/3** of Eq.28 and Eq.31. So the *fractional* quantum Hall effect.

For example, an orbit of 3 × de Broglie wavelength (**n=3**) contains **one** electron.

( As a result, **k= 1**).

As shown in Eq.34, the orbital radius becomes smaller (= **repulsive** force among electrons is stronger ), as the magnetic field (B) is **stronger**.

As a result, **fractional** Hall conductance ( based on small electron's number ) is often seen under **high** magnetic field B ( see Fig.28 ).

*(Fig.31) Diagonal resistance = 0 → each orbit "fits" each other "smoothly".*

As you see Fig.28, Hall plateau appears, when diagonal resistance is zero (= ρ_{xx} = 0 ).

This means each electron is moving **smoothly**, NOT scattered by other electron's orbits.

As you see Fig.31, when each orbital length is 2 × de Broglie wavelength, neighboring orbits are always **attached** smoothly to each other due to **constructive** interference.

The electron "B" touches "A" orbit, and then "C" orbit, while they are rotating.

This model is very **natural**, because we can explain integer and fractional Hall effect using the **common** concept, **NOT** relying on strange "**fractional** charge".

*(Fig.32) "Massless" Dirac fermion is real ? or only illusion ?*

Next we explain how massless Dirac fermion in topological insulator and graphene is described in the current physics.

You will easily find that these concepts are only **mathematical** with **NO** physical images.

*(Eq.40) Original Dirac equation.*

As shown on this page, Dirac equation ( Hamiltonian ) becomes like Eq.40.

If mass of this Dirac fermion is zero, this 4 × 4 matrices can be divided into 2 × 2 matrices, as follows

*(Eq.41) Dirac Hamiltonian ( m = 0 ).*

where Pauli matrices σ are

*(Eq.42)*

Eigenvalues of Eq.43 turn out to be

*(Eq.44) *

where wavenumber "k" is

*(Eq.45) *

*(Fig.33) "Massless" Dirac fermion ? *

The relation of Eq.44 becomes like Fig.33.

So, they argure this means **massless** Dirac fermion like photon.

But as you know, massless Dirac fermion (= electron ) does **NOT** really exist.

Tis is strange.

And you find one of eigenfunctions of Eq.43 becomes

*(Eq.46) *

Substituting Eq.43 and Eq.46 into the left side of Eq.47, this energy becomes **linearly** proportional to wavenumber "k". (= massless !? )

*(Eq.47) *

But there is a **serious** problem in the solution of Eq.46.

When k_{z} = -k, this solution diverges to **infinity**, as follows,

*(Eq.48) *

This is the main reason why massless Dirac fermion is **wrong**.

And this pattern is similar to fictitious magnetic **monopole** of Fig.13.

*(Eq.49) radial direction "k" = 0*

In Eq.9, when arbitrary parameter "R" is wavenumber "k", Berry connection "A" is expressed as Eq.49.

Here we consider the line integral at the **constant** absolute value of |k| on the **equator** ( k_{z} = 0 ) of wavenumber space.

So the partial derivative with respect to "radial k" becomes zero like Eq.49.

*(Eq.50) tangential direction "k"*

We suppose, only in the tangential k_{φ} direction, this Berry connection is not zero.

On the equator, "k_{z}" component is zero, so the wavefunction of Eq.46 becomes

*(Eq.51) *

where

*(Eq.52) *

Substituting Eq.51 into Eq.50, we have

*(Eq.53) *

As a result, the line integral (= Berry curvature, "B" ) of Eq.52 becomes

*(Eq.54) *

Of course, these "A" and "B" are "**fake**" magnetic potentials in "k" ( NOT real ) space.

As shown in Eq.48, this wavefunction φ^{N} diverges to **infinity** at the point of k_{z} = -k.

*(Eq.55) *

So they suddenly introduced new eigenfunction of φ^{S}, as shown in Eq.55.

But this math is very **forcible**.

Because the wavefunction is **discontinuous** on the equator.

*(Eq.56) *

Eq.56 is new wavefunction of south hemisphere ( k_{z} < 0 ).

As you see, Eq.56 **diverges** to infinity, when k_{z} = k.

*(Eq.57) *

On the equator ( k_{z} = 0), this wavefunction becomes like Eq.57.

As you compare Eq.51 and Eq.57, you find these wavefunctions are discontinuous on the k_{z} = 0 line.

*(Eq.58) *

In the same way, the line integral of Eq.57 becomes -π.

As a result, magnetic charge (= monopole ) is the sum of Eq.54 and Eq.58,

*(Eq.59) *

Eq.59 is "monopole" in masless Dirac fermion.

*(Fig.33)*

But this method is very **unreasonable** also from the viewpoint of mathematics.

As you see Fig.33, it is strange that vector potentials "A" become different between clockwise and counterclockwise routes.

*(Fig.34) Barry phase is only math trick. NOT real.*

As I said, Berry phase is very **abstract** math concept.

The current quantum mechanics NEVER try to say what this phase **really** is (= "Shut up and calculate !" ).

*(Eq.60) *

As I said, Berry curvature (= B ) is the curl of Berry connection (= "A" ) like magnetic field ( see Eq.9' ).

Hamiltonian and eigenfunction are

*(Eq.61) *

Based on Eq.60 and Eq.61, we can get

*(Eq.62) *

About the detailed calculation method, see this page.

*(Eq.63) *

On this page, we get the Hall conductance of Eq.63 using Berry curvature of Eq.62.

*(Eq.64) *

In Eq.63, the value of Eq.64 is called "**Chern** number".

When this Chern number becomes an integer, Hall conductance of Eq.63 becomes quantized (= integer Hall effect ).

*(Fig.35) Artificial definition of different Barry connection.*

But like other magnetic monopole, if we try to calculate Eq.64 in all "k" space surface of Brillouin zone, the total Chern number always becomes **zero**.

To avoid this cancellation, again, they define **discontinuous** "A", as shown in Fig.35.

In Fig.35, Berry phase "A" suddenly changes on the borderline between A^{II} and A^{I}.

This is clearly one of **artificial** tricks with **NO** reasonable reasons.

*(Eq.65) *

Here we use Berry connection with parameter R = k (= wavenumber ) in Eq.9.

Furthermore, they define another **artificial** rule of

*(Eq.66) *

From Eq.64, Eq.65 and Eq.66, total Chern number becomes

*(Eq.67) *

As a result,

*(Eq.68) *

When this phase is an integer (= n ) times 2π in one orbit,

*(Eq.69) *

Hall conductance of Eq.63 becomes

*(Eq.70) *

So they insist Berry phase can explain integer quantum Hall effect.

But as you see Eq.65, this parameter R (= k ) is the **function** of the time "t".

As a result, as time goes by, this Chern number **increases** to infinity ! ( ← NOT constant value. )

*(Eq.71) *

So, Hall conductance σ increases to **infinity** with time ? This is strange.

*(Eq.72) *

And the definition of Eq.66 is artificial.

Because, wavefunction of φ_{II} ( given using θ(t) ) itself is **changing** with time ( t, k ).

*(Fig.36) Half-integer QHE in graphen.*

It is known that graphene ( carbon ) monolayer shows special type of quantum Hall effect.

As shown in Fig.36 and this site, filling factor (= n ) becomes ±2, ±6, ±10 ...

But the detailed mechanism of this graphene is still **unknown**.

But later, as shown on this site and this site, **other** components such as 0, ±1, ±4 and **fractions** were found.

So the relation of Fig.36 is **NOT** correct.

*(Fig.37) Mechanism of half-integer QHE in graphene.*

Even in 2-dimensional electron system, there is some **thickness** in the electron's layer.

Due to the graphene monolayer ( electron ), other electrons are **repelled** near graphene layer.

This is the reason why the first filling factor ( related to electron's **density** ) is n = ±2, and then increases like n = ±6, ±10, I think.

*(Fig.38) Massless Dirac particle also in graphene ?*

It is known that honeycomb lattice graphene also has **massless** Dirac fermions, as shown on this site and this site.

Again, this massless fermion is only mathematical concept with **NO** physical entity.

At the Dirac cone, there are two **gapless** energy between conduction and valence bands.

About the detailed calculation, see this page and this section.

*(Fig.39) Angle-resolved photoemission spectroscopy ( ARPES )*

To measure the band structure and massless Dirac fermion, we have to depend on ARPES (= angle-resolved photoemission spectroscopy ).

The mechanism of this ARPES is very simple.

By illuminating the surface of some material, photoelectron is ejected.

After measuring energy and each component of electron's momentum, they determine band structure and dispersions (= relation between wavenumebr and energy ).

In Fig.39, ħω is the energy of this light, and "K" is wavenumber (= momentum ) **after** going out of material.

*(Fig.40) It is impossible to know the correct momentum inside material with ARPES. *

Of course, we want to know the relation between momentum (= small "k" ) and energy (= E_{i} ) **inside** material.

But it is basically **impossible** to know the precise momentum inside material only from ejected photoelectrons.

But they are **mistaking** the momentums of photoelectron **outside** for the momentum **inside** material.

This is the most serious **defects** in ARPES experiment, which means massless Dirac fermion is very **doubtful**.

*(Eq.73) Momentum (= K ), energy (= E ) outside.*

Eq.73 is momentum and energy of photoelectrons **outside** the material.

These components are parallel or perpendicular to the surface of the material.

*(Eq.74) *

Surprisingly, they **jump** to a conclution that the ( parallel ) momentums inside and outside are completely the **same**.

This is very strange.

because, it is quite natural that momentum informations are completely **different** between free electrons and **bound** electrons of lower potential energy.

This means when the electron is illuminated, original momentum of electron completely **changes**.

From Eq.73 and Eq.74, they get

*(Eq.75) *

where W is work function, which must be determined only from experiments.

*(Eq.76) *

V_{0} is also **unknown** parameter, which must be estimated from experiments.

Of course, the precondition of Eq.74 is **wrong**, this means all these results based on ARPES are all **meaningless**.

In fact, as shown on this site (p.49), the experimental results are a little **different** from theoretical prediction.

( ex. to be precise, Dirac cone is **NOT** gapless. )

*(Fig.41) *

In this paper ( Science 318 766, 2007 ), they insist spin Hall effect in topological insulator (HgTe) was observed.

But this experiment is completely **different** from usual spin Hall effect.

Of course, strange spin **cannot** be measured directly in any experiments.

*(Eq.77) *

In this experiment, they observe the conductance of usual **integer** quantum Hall effect, as shown in Eq.77.

They insist, electric current is **bidirectional** on the surface of topological insulator.

So "2" of Eq.77 is theoretical value.

But in this paper, this conductance is **NOT** constant.

Only under very **special** condition, Eq.77 is gotten.

So only from this experiment, we **cannot** prove spin Hall effect at all.

2014/3/20 updated. Feel free to link to this site.