Berry phase effects on electronic properties

The Berry connection is not gauge-independent.

“This is not surprising because the Berry curvature is a gauge-independent quantity and the Berry connection is not.”

In the presence of an electric field, an electron can acquire an anomalous velocity proportional to the Berry curvature of the band.

“In the presence of an electric field, an electron can acquire an anomalous velocity proportional to the Berry curvature of the band (Chang and Niu, 1995, 1996; Sundaram and Niu, 1999).”

• cites Chang and Niu, 1995, 1996Not checked yet

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• cites Sundaram and Niu, 1999Not checked yet

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The adiabatic current induced by a time-dependent perturbation in a band is equal to the q-integral of the Berry curvature Ω_nqt.

“We have thus derived the remarkable result that the adiabatic current induced by a time-dependent perturbation in a band is equal to the q -integral of the Berry curvature Ω n qt (Thouless, 1983).”

• cites Thouless, 1983Not checked yet

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The path C must be closed to make γ_n a gauge-invariant quantity with physical significance.

“We emphasize that the path C must be closed to make γ n a gauge-invariant quantity with physical significance.”

The quantization of the Chern number is always true as long as the Hamiltonian is periodic in both parameters.

“Note that in our proof, we made no reference to the original physical system; the quantization of the Chern number is always true as long as the Hamiltonian is periodic in both parameters.”

The Berry curvature of energy bands is an intrinsic property of the band structure because it only depends on the wave function.

“The Berry curvature Ω n ( q ) is an intrinsic property of the band structure because it only depends on the wave function.”

The Berry curvature is gauge-independent.

“This is not surprising because the Berry curvature is a gauge-independent quantity and the Berry connection is not.”

The Berry curvature itself is a local gauge-invariant quantity.

“In fact, once we have introduced the concept of the Berry curvature, a closed loop is not necessary because the Berry curvature itself is a local gauge-invariant quantity.”

The polarization difference between two states is given by the integrated bulk current as the system adiabatically evolves from the initial state at t=0 to the final state at t=T.

“The above equation can be interpreted in the following way: The polarization difference between two states is given by the integrated bulk current as the system adiabatically evolves from the initial state at t = 0 to the final state at t = T .”

In the Rice-Mele model with ∆=0, the polarization difference between P(δ) and P(-δ) is ea/2.

“Therefore the polarization difference between P ( δ ) and P ( -δ ) is ea/ 2.”

The criterion for c_n to be nonzero is that a degeneracy point must occur somewhere inside the torus as one varies q, R1 and R2.

“Thus, the criterion for c n to be nonzero is that a degeneracy point must occur somewhere inside the torus as one varies q , R 1 and R 2 .”

In a one-dimensional system with inversion symmetry, the polarization density P is either 0 or ea/2 modulo ea.

“Therefore P is either 0 or ea/ 2 (modulo ea ). Any other value of P will break the inversion symmetry.”

The Berry phase over a torus is quantized in the unit of 2π.

“Here we prove that it is also true in the case of a torus.”

The polarization obtained from the two-point formula is determined up to an uncertainty quantum.

“However, a price must be paid to use the two-point formula, namely, the polarization in Eq. (2.28) is determined up to an uncertainty quantum.”

In a one-dimensional system with inversion symmetry, Zak's phase can only take the value 0 or π modulo 2π.

“Consequently, Zak's phase can only take the value 0 or π (modulo 2 π ).”

If R(t) is a smooth function of t, then R must have at least two components for its trajectory to trace out a circle on the torus.

“If R ( t ) is a smooth function of t , as it is usually the case for physical quantities, then R must have at least two components, say R 1 and R 2 . Otherwise the trajectory of R ( t ) cannot trace out a circle on the torus [see Fig. 1(a)].”

The transported particle number over a time cycle is quantized and is an integer called the first Chern number.

“Therefore the transported particle number c , given in Eq. (2.11), must be quantized. This integer is called the first Chern number, which characterizes the topological structure of the mapping from the parameter space ( q, t ) to the Bloch states | u ( q, t ) 〉 .”

The calculated electron density using Eq. (6.18) and the numerical result from direct diagonalization of the tight-binding Hamiltonian are virtually indistinguishable.

“Figure 11 shows the calculated electron density using Eq. (6.18) together with numerical result obtained by direct diagonalization of the tight-binding Hamiltonian. These two results are virtually indistinguishable in the plot”

Since the split magnetic subband is narrower and flatter than the original Bloch band, the electron group velocity is small.

“Furthermore, since the split magnetic subband is narrower and flatter than the original Bloch band, the electron group velocity is small.”

The states on the opposite sides of the k1 boundaries have the same phase.

“That is, the states on the opposite sides of the k 1 boundaries have the same phase.”

The size of a real-space hyperorbit may be very large if phase coherence can be maintained during the circulation, since it is proportional to the inverse of the residual magnetic field δB.

“The size of a real-space hyperorbit may be very large (if phase coherence can be maintained during the circulation) since it is proportional to the inverse of the residual magnetic field δB .”

The area of the largest hyperorbit should be equal to or slightly smaller than the area of the MBZ (2π/a)^2/q, assuming δB << B0 so that the number of hyperorbits is large.

“The area of the largest hyperorbit should be equal to or slightly smaller (assuming δB ≪ B 0 so that the number of hyperorbits is large) than the area of the MBZ (2 π/a ) 2 /q (see Fig. 13(b)).”

In the case of the MBB, the magnetic field B0 changes the band structure itself.

“In the case of the MBB, the magnetic field B 0 changes the band structure itself.”

The r-orbit is simply the k-orbit rotated by π/2 and linearly scaled by the factor hbar/eδB.

“Similar to the Bloch band case, it is not difficult to see from Eq. (8.23) that the r -orbit is simply the k -orbit rotated by π/ 2 and (linearly) scaled by the factor ¯ h/eδB .”

The degeneracy of the original Landau level is Φ/φ0 = Np/q, where Φ=Nφ is the total magnetic flux through the two-dimensional sample.

“The degeneracy of the original LL is Φ /φ 0 = Np/q , where Φ = Nφ is the total magnetic flux through the two dimensional sample.”

If the magnetic flux φ enclosed by a plaquette is (p/q)φ0, where p and q are co-prime integers, then T^q_{a1} commutes with T_{a2}.

“If the magnetic flux φ enclosed by a plaquette is ( p/q ) φ 0 , where p and q are co-prime integers, then T q a 1 would commute with T a 2 (see Fig. 12).”

The magnetic translation operator T_a defined in the text commutes with the Hamiltonian H.

“The operator T a being defined this way has the desired property that [ H,T a ]=0.”

T_a1 commutes with T_a2 only if the magnetic flux φ is an integer multiple of the flux quantum φ0 = e/h.

“T a 1 commutes with T a 2 only if the flux φ is an integer multiple of the flux quantum φ 0 = e/h .”

The k-orbit moves along a path with constant E(k)=E0(k)-M(k)·δB, which is the magnetic Bloch band energy shifted by the magnetization energy.

“This determines the k -orbit moving along a path with constant E ( k ) = E 0 ( k ) -M ( k ) · δ B , which is the magnetic Bloch band energy shifted by the magnetization energy.”

Since the magnetic unit cell is q times larger than the usual unit cell, the magnetic Brillouin zone (MBZ) has to be q times smaller.

“Since the magnetic unit cell is q times larger than the usual unit cell, the magnetic Brillouin zone (MBZ) has to be q times smaller.”

Therefore, the MBZ has a q-fold degeneracy along the b2 direction.

“Therefore, the MBZ has a q -fold degeneracy along the b 2 direction.”

The number of hyperorbits should be |1/(qδφ)+σ_H|, where δφ≡δB a^2/φ0 is the residual flux per plaquette.

“Therefore, it is not difficult to see that the number of hyperorbits should be | 1 / ( qδφ ) + σ H | , where δφ ≡ δBa 2 /φ 0 is the residual flux per plaquette.”

Magnetic translations along different directions usually do not commute.

“Unlike usual translation operators, magnetic translations along different directions usually do not commute.”

After splitting, each magnetic Bloch band again has only N/q states, the number of states in a MBZ.

“Therefore, after splitting, each MBB again has only N/q states, the number of states in a MBZ.”

When δB is not zero, there exists a similar circulating motion in the MBB, called a hyperorbit.

“As a result, when δ B is not zero, there exists similar circulating motion in the MBB. This type of orbit will be called 'hyperorbit'.”

In the example, the MBZ is folded back q times along the b1 direction.

“Then, in this example, the MBZ is folded back q times along the b 1 direction.”

For such an orbit, the Berry phase correction Γ/2π in Eq. (7.3) is very close to the integer Hall conductivity σ_H of the MBB.

“For such an orbit, the Berry phase correction Γ / 2 π in Eq. (7.3) is very close to the integer Hall conductivity σ H of the MBB.”

As a result of imposing the parallel-transport condition, the phases of any two states in the MBZ have a definite relation.

“As a result, the phases of any two states in the MBZ have a definite relation.”

After quantization, there can only be a finite number of hyperorbits in the MBZ.

“After the quantization, there can only be a finite number of hyperorbits in the MBZ.”

Except for the extension to multiple components, the form of the magnetic moment remains the same as its Abelian counterpart.

“Except for the extension to multiple components, the form of the magnetic moment remains the same as its Abelian counterpart (see Eq. (4.6)).”

Because of the spin-orbit coupling between the electron and the spinless impurity, the scattering amplitude is not symmetric with respect to the transverse direction.

“Because of the spin-orbit coupling between the electron and the (spinless) impurity, the scattering amplitude is not symmetric with respect to the transverse direction.”

The Berry curvature emerges when one restricts the whole Hilbert space to a particular (HH or LH) subspace.

“The Berry curvature emerges when one restricts the whole Hilbert space to a particular (HH or LH) subspace.”

Compared to the Abelian case in Eq. (5.8), the kdot equation also has the electric force and the Lorentz force.

“Compared to the Abelian case in Eq. (5.8), the ˙ k c -equation also has the electric force and the Lorentz force.”

The anomalous velocity in Eq. (9.7) remains essentially the same, with the Abelian Berry curvature replaced by the non-Abelian one.

“Also, the anomalous velocity in Eq. (9.7) remains essentially the same. One only needs to replace the Abelian Berry curvature with the non-Abelian one.”

These are magnetic energy subbands at a finer energy scale compared to the original MBB.

“These are the magnetic energy subbands at a finer energy scale compared to the original MBB.”

The Hall conductivity is σ_close_r = eρ/δB.

“Therefore, the Hall conductivity is σ close r = eρ/δB .”

For a closed orbit, the motion is a cyclotron motion superimposed with a drift along the E × δB direction.

“For a closed orbit, this is just a cyclotron motion superimposed with a drift along the E × δ B direction.”

After time average, the cyclotron motion does not contribute to a net transport.

“After time average, the former does not contribute to a net transport.”

Same as the Abelian case, the wavepacket energy has three terms: the Bloch energy, the electrostatic energy, and the magnetization energy.

“Same as the Abelian case, it has three terms : the Bloch energy, the electrostatic energy, and the magnetization energy.”

The sum of these integers equals the original Hall conductivity: σ_H = ∑_r σ_r.

“The sum of these integers should be equal to the original Hall conductivity: σ H = ∑ r σ r .”

For lattices with square or triangular symmetry, there is one, and only one, nesting (open) hyperorbit in the MBZ.

“For lattices with square or triangular symmetry, there is one, and only one, nesting (open) hyperorbit in the MBZ (for example, see the diamond-shaped energy contour in Fig. 13(b)).”

The dipole-energy term eE·R originates from the shift between the charge center r_c and the center of the canonical variable r.

“The dipole-energy term e E · R is originated from the shift between the charge center r c and the center of the canonical variable r .”

The form of the Hamiltonian, especially the spin-orbit term and Yafet term, is clearly gauge dependent because of the gauge-dependent Berry connection.

“First, the form of the Hamiltonian, especially the spin-orbit term and Yafet term, is clearly gauge dependent because of the gaugedependent Berry connection.”

The semiclassical theory based on the variables F and M is gauge independent.

“Third, the semiclassical theory based on the variables F and M , on the other hand, are gauge independent.”

Compared to the semiclassical Hamiltonian in Eq. (9.9), the quantum Hamiltonian has two additional terms from the Taylor expansion.

“Compared to the semiclassical Hamiltonian in Eq. (9.9), the quantum Hamiltonian has two additional terms from the Taylor expansion.”

To the lowest order of k, the rotating wavepacket in the conduction band has a magnetic moment.

“To the lowest order of k , it has the magnetic moment,”

If the first Chern number (or its non-Abelian generalization) is not zero, one cannot choose a global gauge in which R is smooth everywhere in the Brillouin zone.

“However, if the first Chern number (or its non-Abelian generalization) is not zero, one cannot choose a global gauge in which R is smooth everywhere in the Brillouin zone.”

If all energy levels are included, the total Berry curvature vanishes for each value of R.

“In fact, if all energy levels are included, it follows from Eq. (1.13) that the total Berry curvature vanishes for each value of R ,”

In the absence of degeneracy, an eigenstate evolving adiabatically around a closed loop returns to itself with a phase difference equal to the time integral of the energy plus an extra Berry phase.

“In the absence of degeneracy, the eigenstate will surely come back to itself when finishing the loop, but there will be a phase difference equal to the time integral of the energy (divided by ¯ h ) plus an extra which is later commonly called the Berry phase.”

• cites Berry, 1984Not checked yet

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The Berry phase is physical.

“This property makes the Berry phase physical, and the early experimental studies were focused on measuring it directly through interference phenomena.”

It is necessary and sufficient to include the Berry phases and gradient energy corrections in addition to the energy dispersions in order to account for all phenomena up to first order in the fields.

“It is necessary and sufficient to include the Berry phases and gradient energy corrections in addition to the energy dispersions in order to account all phenomena up to first order in the fields.”

According to the quantum adiabatic theorem, a system initially in an eigenstate will stay as an instantaneous eigenstate of the Hamiltonian throughout the process.

“According to the quantum adiabatic theorem (Kato, 1950; Messiah, 1962), a system initially in one of its eigenstates | n ( R (0)) 〉 will stay as an instantaneous eigenstate of the Hamiltonian H ( R ( t )) throughout the process.”

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• cites Messiah, 1962Not checked yet

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Based on the Berry phase formula, polarization-related properties can now be routinely calculated using first-principles methods with a typical precision of density functional theory.

“Based on the Berry phase formula, one can now routinely calculate polarization related properties using first principles methods, with a typical precision of the density functional theory.”

By including a side-jump term into the usual Boltzmann equation, the semiclassical theory can reproduce all the intricacies of linear response theory in weakly disordered systems.

“Moreover, by including a side-jump term (which is itself a kind of Berry phase effect) into the usual Boltzmann equation and associated transport during collisions as well as anomalous transport between collisions, the semiclassical theory can also reproduce all the intricacies of linear response theory in weakly disordered systems (Sinitsyn, 2008).”

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The Berry curvature can be regarded as the result of the residual interaction of projected-out energy levels.

“In view of Eq. (1.13), the Berry curvature can be regarded as the result of the 'residual' interaction of those projected-out energy levels.”

Spin-orbit coupling and various Yafet terms are explained by the generalized Peierls substitution.

“Spin-orbit coupling and various mysterious Yafet terms are all found a simple explanation from this generalized Peierls substitution.”

The Berry phase of the electronic wave function is responsible for a spectrum of phenomena, including ferroelectricity, orbital magnetism, various (quantum/anomalous/spin) Hall effects, and quantum charge pumping.

“is responsible for a spectrum of phenomena, such as ferroelectricity, orbital magnetism, various (quantum/anomalous/spin) Hall effects, and quantum charge pumping”

The Berry phase of the electronic wave function can have a profound effect on material properties.

“the Berry phase of the electronic wave function can have a profound effect on material properties”

The Berry phase can be written as an integral of the Berry curvature over a surface suspending the loop.

“Indeed, Berry himself showed that one can write the Berry phase as an integral of a field, which we now call as the Berry curvature, over a surface suspending the loop.”

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The Berry phase is gauge invariant up to an unessential integer multiple of 2π when the eigen-wavefunction is kept single valued over the loop.

“The Berry phase is unchanged (up to unessential integer multiple of 2 π ) by such a phase factor, provided the eigen-wavefunction is kept to be single valued over the loop.”

A generalized Peierls substitution relating physical variables to canonical variables turns the semiclassical dynamics into a full quantum theory valid to first order in the fields.

“Finally, one can execute a generalized Peierls substitution relating the physical variables to the canonical variables, turning the semiclassical dynamics into a full quantum theory valid to first order in the fields (Chang and Niu, 2008).”

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The quantized Hall conductivity in two dimensions is robust against many-body interactions and disorder.

“Niu et al. (1985) further showed that the quantized Hall conductivity in two-dimensions is robust against many-body interactions and disorder.”

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The usual description of Bloch electron dynamics under an electric field, in which the electron velocity is simply given by the band energy derivative, is incomplete.

“Through recent progress on the semiclassical dynamics of Bloch electrons, it has been made increasingly clear that this description is incomplete.”

The concepts of gauge field and geometric phase can still be applied to open mesoscopic systems described by the scattering matrix.

“Although the physical description of these open systems are dramatically different from the closed ones, the concepts of gauge field and geometric phase can still be applied.”

If the potential shifts one spatial period in the time cycle, the particle transport equals the number of filled Bloch bands (double if spin degeneracy is counted).

“If the potential shifts one spatial period in the time cycle, the particle transport should be equal to the number of filled Bloch bands (double if the spin degeneracy is counted).”

The Hall conductivity σ_xy is the Chern number in units of e^2/h.

“Here σ xy is just the Chern number in the units of e 2 /h , i.e.,”

In systems with either broken time-reversal or inversion symmetries, a proper description requires the full velocity formula including the Berry curvature.

“However, in systems with either broken time-reversal or inversion symmetries, their proper description requires the use of the full velocity formula (3.6).”

If the Fermi energy lies in a gap, the current j(κ) is insensitive to the boundary condition specified by κ.

“The key step is achieved by realizing that if the Fermi energy lies in a gap, then the current j ( κ ) should be insensitive to the boundary condition specified by κ (Niu and Thouless, 1984; Thouless, 1981).”

• cites Niu and Thouless, 1984Not checked yet

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• cites Thouless, 1981Not checked yet

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The Berry curvature is a more fundamental quantity than the Berry phase.

“In this sense, the Berry curvature is a more fundamental quantity than the Berry phase.”

The Hall conductivity is quantized for a two-dimensional band insulator of non-interacting electrons.

“Therefore the Hall conductivity is quantized for a twodimensional band insulator of non-interacting electrons.”

The electron velocity in a Bloch state under an electric field has, in addition to the usual band dispersion contribution, an anomalous velocity term.

“We can see that, in addition to the usual band dispersion contribution, an extra term previously known as an anomalous velocity also contributes to v n ( k ).”

The Berry curvature is nonzero in a wide range of materials, in particular crystals with broken time-reversal or inversion symmetry.

“It is nonzero in a wide range of materials, in particular, crystals with broken time-reversal or inversion symmetry.”

This anomalous velocity is responsible for a number of transport phenomena, in particular various Hall effects.

“This anomalous velocity is responsible for a number of transport phenomena, in particular various Hall effects, which we study in this section.”

The polarization may be regarded as a multi-valued quantity with each value differing by the quantum.

“Because of this uncertainty quantum, the polarization may be regarded as a multi-valued quantity with each value differed by the quantum.”

The Hall conductivity is quantized even in the presence of many-body interaction and disorder.

“Therefore the Hall conductivity is quantized even in the presence of many-body interaction and disorder.”

In the presence of inversion symmetry, Zak's phase γ_n is either 0 or π.

“By analyzing the symmetry properties of Wannier functions (Kohn, 1959) of a one-dimensional crystal, Zak (1989a) showed that γ n is either 0 or π in the presence of inversion symmetry; a simple argument is given in Sec. II.C.”

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• cites Zak (1989a)Not checked yet

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In a strong magnetic field, the Hall conductivity of a two-dimensional electron gas is exactly quantized in units of e^2/h.

“They found that in a strong magnetic field the Hall conductivity of a two-dimensional electron gas is exactly quantized in the units of e 2 /h .”

• cites Klitzing et al. (1980)Not checked yet

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If the crystal lacks inversion symmetry, Zak's phase γ_n can assume any value.

“If the crystal lacks inversion symmetry, γ n can assume any value.”

• cites Zak (1989a)Not checked yet

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For a two-dimensional band insulator, the Hall conductivity is given by the integral of the Berry curvature over the entire Brillouin zone.

“It follows from Eq. (3.6) that the Hall conductivity of the system is given by”

In graphene with broken inversion symmetry, the Berry curvature at valleys K1 and K2 has opposite signs due to time-reversal symmetry.

“The Berry curvature at valley K 1 and K 2 have opposite signs due to time-reversal symmetry.”

• cites Xiao et al. , 2007bNot checked yet

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Electric polarization can be regarded as 'unquantized' adiabatic particle transport.

“This description implies a time-dependent Hamiltonian H ( t ), and the electric polarization can be regarded as 'unquantized' adiabatic particle transport.”

Information of the Berry curvature is essential in a proper description of the dynamics of Bloch electrons.

“It is now well recognized that information of the Berry curvature is essential in a proper description of the dynamics of Bloch electrons, which has various effects on transport and thermodynamic properties of crystals.”

The existence of a bulk energy gap guarantees that the Hall conductivity remains unchanged after thermodynamic averaging.

“The existence of a bulk energy gap guarantees that the Hall conductivity remains unchanged after thermodynamic averaging, which is given by”

For crystals with simultaneous time-reversal and spatial inversion symmetry, the Berry curvature vanishes identically throughout the Brillouin zone.

“Therefore, for crystals with simultaneous time-reversal and spatial inversion symmetry the Berry curvature vanish identically throughout the Brillouin zone.”

In practice, effective Hamiltonians obtained by projecting a full Hamiltonian are accompanied by some residual Berry curvature.

“However, in practice one usually deals with effective Hamiltonians which are obtained through some projection process of the full Hamiltonian. Therefore there will always be some 'residual' Berry curvature accompanying the effective Hamiltonian.”

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The generic two-level Hamiltonian describes a number of physical systems in condensed matter physics for which the Berry phase effect has been discussed.

“Despite its simple form, the above Hamiltonian describes a number of physical systems in condensed matter physics for which the Berry phase effect has been discussed.”

• cites Culcer et al. , 2003; Liu et al. , 2008Not checked yet

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• cites Rice and Mele, 1982; Su et al. , 1979Not checked yet

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• cites Onoda et al. , 2004b; Vanderbilt and King-Smith, 1993Not checked yet

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As the gap approaches zero, the Berry phase acquired by an electron during one circle around the valley becomes exactly ±π.

“We note that as the gap approaches zero, the Berry phase acquired by an electron during one circle around the valley becomes exactly ± π .”

The number of transported electrons can be related to the number of monopoles, which are degeneracy points in the parameter space.

“Therefore the number of transported electrons can be related to the number of monopoles, which are degeneracy points in the parameter space.”

It is always the change of the polarization that has been measured.

“The above interpretation is also consistent with experiments, as it is always the change of the polarization that has been measured (Resta and Vanderbilt, 2007).”

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The degeneracy points of a two-level system act as sources and drains of the Berry curvature flux.

“Therefore the degeneracy points act as sources and drains of the Berry curvature flux.”

In the general case with many-body interactions and disorder, the quantization of particle transport is still valid as long as the system remains an insulator during the whole process.

“Niu and Thouless (1984) studied this problem and showed that in the general case the quantization of particle transport is still valid as long as the system remains an insulator during the whole process.”

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Unless the system has both time-reversal and inversion symmetry, the orbital moment m(q) is generally nonzero.

“Therefore unless the system has both time-reversal and inversion symmetry, m ( q ) is in general nonzero.”

The size of the wave packet has a nonzero lower bound because it is constructed using an incomplete basis of Bloch functions.

“In fact, since it is constructed using an incomplete basis of the Bloch functions, the size of the wave packet has a nonzero lower bound (Marzari and Vanderbilt, 1997).”

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Under symmetry operations, the orbital moment transforms exactly like the Berry curvature.

“Under symmetry operations, the orbital moment transforms exactly like the Berry curvature.”

Upon application of a magnetic field, the orbital moment couples to the field through a Zeeman-like term -m(q)·B.

“For example, upon the application of a magnetic field, the orbital moment will couple to the field through a Zeeman-like term -m ( q ) · B .”

The integration of the Berry curvature of a full band is ±π for the upper and lower bands.

“One can verify that the integration of the Berry curvature of a full band, 2 π ∫ ∞ 0 qdq Ω ± , is ± π for the upper and lower bands, respectively.”

The Hall conductivity can be written as the Berry phase of quasiparticles on the Fermi surface, up to a multiple of 2π.

“This issue was first raised by Haldane (2004), and he showed that the Hall conductivity can be written, in the units of e 2 / 2 πh , as the Berry phase of quasiparticles on the Fermi surface, up to a multiple of 2 π .”

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The Berry phase around the Dirac points becomes exactly ±π.

“Meanwhile, The Berry phase around the Dirac points becomes exactly ± π (also see Sec. VII.C).”

In metallic systems, electron scattering can be important in transport phenomena.

“The above discussion does not take into account the fact that, unlike insulators, in metallic systems electron scattering can be important in transport phenomena.”

The Berry curvatures in the two valleys have opposite sign, as required by time-reversal symmetry.

“Note that the Berry curvatures in two valleys have opposite sign, as required by time-reversal symmetry.”

The Berry phase on the Fermi surface should be added as a topological ingredient to the Landau Fermi liquid theory.

“This observation suggests that the Berry phase on the Fermi surface should be added as a topological ingredient to the Landau Fermi liquid theory.”

For the graphene model with broken inversion symmetry, the orbital moment of the conduction band is given by Eq. (4.7).

“For the graphene model with broken-inversion symmetry, discussed in Sec. III.E, the orbital moment of the conduction band is given by (Xiao et al. , 2007b)”

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Two contributions to the anomalous Hall effect arise due to scattering: skew scattering and side jump.

“Two contributions to the anomalous Hall effect arises due to scattering: (i) the skew scattering that refers to the asymmetric scattering amplitude with respect to the scattering angle between the incoming and outgoing electron waves (Smit, 1958), and (ii) the side jump which is a sudden shift of the electron coordinates during scattering (Berger, 1970).”

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• cites Berger, 1970Not checked yet

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The Hall conductivity is resonantly enhanced only when -∆ < ε_F < ∆.

“It is only when -∆ &lt; ε F &lt; ∆, the Hall conductivity is resonantly enhanced (Onoda et al. , 2006b).”

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In both semiclassical and full quantum mechanical approaches, the Berry-phase supported intrinsic contribution to the anomalous Hall effect has been firmly established.

“In both approaches, the Berryphase supported intrinsic contribution to the anomalous Hall effect has been firmly established.”

• cites Sinitsyn, 2008Not checked yet

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• cites Onoda et al. , 2006bNot checked yet

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• cites Sinitsyn et al. , 2007Not checked yet

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The magnetic dipole e r × p is not defined in a periodic system.

“The main difficulty of calculating the orbital magnetization is exactly the same one we faced when calculating the electric polarization: the magnetic dipole e r × p is not defined in a periodic system.”

The intrinsic contribution to the anomalous Hall effect can be regarded as an unquantized version of the quantum Hall effect.

“The intrinsic contribution to the anomalous Hall effect can be regarded as an 'unquantized' version of the quantum Hall effect.”

There was no proper way to calculate orbital magnetization until recently when the Berry phase theory of orbital magnetization was developed.

“In fact, there was no proper way to calculate this quantity until recently when the Berry phase theory of orbital magnetization is developed (Shi et al. , 2007; Thonhauser et al. , 2005; Xiao et al. , 2005).”

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• cites Thonhauser et al. , 2005Not checked yet

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• cites Xiao et al. , 2005Not checked yet

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The scattering-independent intrinsic contribution to the anomalous Hall effect comes from the Berry-phase supported anomalous velocity.

“It was later realized that the scattering-independent intrinsic contribution comes from the Berry-phase supported anomalous velocity.”

Broken inversion symmetry allows a valley Hall effect.

“It is this broken inversion symmetry that allows a valley Hall effect.”

The orbital moment behaves exactly like the electron spin.

“This orbital moment behaves exactly like the electron spin.”

Zhou et al. reported observation of a band gap opening in epitaxial graphene, attributed to inversion symmetry breaking by the substrate potential.

“Zhou et al. (2007) have recently reported the observation of a band gap opening in epitaxial graphene, attributed to the inversion symmetry breaking by the substrate potential.”

• cites Zhou et al. (2007)Not checked yet

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Intrinsic graphene has two degenerate and inequivalent Dirac points at the corners of the Brillouin zone, where conduction and valence bands touch, forming a valley structure.

“The band structure of intrinsic graphene has two degenerate and inequivalent Dirac points at the corners of the Brillouin zone, where the conduction and valance bands touch each other, forming a valley structure.”

A wave packet may possess a self-rotation around its center of mass, which gives rise to an orbital magnetic moment.

“Therefore, a wave packet may possess a self-rotation around its center of mass, which will in turn give rise to an orbital magnetic moment.”

The orbital moment is an intrinsic property of the band; its expression does not depend on the actual shape and size of the wave packet, only on the Bloch functions.

“We emphasize that the orbital moment is an intrinsic property of the band. Its final expression, Eq. (4.6), does not depend on the actual shape and size of the wave packet, and only depends on the Bloch functions.”

The valley Hall conductivity can be nonzero when inversion symmetry is broken, even if time reversal symmetry remains.

“Therefore, the valley Hall conductivity can be nonzero when the inversion symmetry is broken, even if time reversal symmetry remains.”

The Hall conductivity is the sum of intrinsic, skew scattering, and side jump contributions.

“The Hall conductivity is the sum of different contributions”

Three mechanisms have been identified for the anomalous Hall effect: the intrinsic contribution and the extrinsic contributions from skew and side-jump scattering.

“In the past, three mechanisms have been identified: the intrinsic contribution (Karplus and Luttinger, 1954; Luttinger, 1958), and the extrinsic contributions from the skew (Smit, 1958) and side-jump scattering (Berger, 1970).”

• cites Karplus and Luttinger, 1954Not checked yet

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• cites Luttinger, 1958Not checked yet

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• cites Smit, 1958Not checked yet

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In dirty samples and at high temperatures, σ_sk becomes small compared to both σ_in and σ_sj.

“On the other hand, in dirty samples and at high temperatures, σ sk H becomes small compared to both σ in H and σ sj H .”

Shindou and Balents derived an effective Boltzmann equation for quasiparticles on the Fermi surface using the Keldysh formalism, where the Berry phase of the Fermi surface is defined in terms of quasiparticle Green functions.

“On the theory side, Shindou and Balents (2006, 2008) derived an effective Boltzmann equation for quasiparticles on the Fermi surface using the Keldysh formalism, where the Berry phase of the Fermi surface is defined in terms of the quasiparticle Green functions”

• cites Shindou and Balents (2006, 2008)Not checked yet

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The intrinsic Hall conductivity is also a Fermi surface property.

“Therefore the intrinsic Hall conductivity is also a Fermi surface property.”

For a wave packet, the magnetic dipole is not a problem because it is localized in space.

“For a wave packet this is not a problem because it is localized in space.”

The wave packet has a finite spread in real space.

“The wave packet, unlike a classical point particle, has a finite spread in real space.”

If one constructs a wave packet using only the positive energy states of the Dirac Hamiltonian, its orbital moment in the non-relativistic limit is exactly the Bohr magneton.

“If one construct a wave packet using only the positive energy states (the electron branch) of the Dirac Hamiltonian, its orbital moment in the non-relativistic limit is exactly the Bohr magneton (Sec. IX).”

Valley transport in systems with broken inversion symmetry is a very general phenomenon.

“As we can see, the valley transport in systems with broken inversion symmetry is a very general phenomenon.”

Orbital moments in different valleys have opposite signs, as required by time-reversal symmetry.

“So orbital moments in different valleys have opposite signs, as required by time-reversal symmetry.”

When the Fermi energy is in the gap region, the intrinsic Hall conductivity reaches its maximum value of about -e^2/2h.

“As we can see, when the Fermi energy ε F is in the gap region, the Hall conductivity reaches its maximum value (about -e 2 / 2 h ).”

As ∆ goes to zero, the Berry curvature vanishes everywhere except at the Dirac points where it diverges.

“We remark that as ∆ goes to zero, the Berry curvature vanishes everywhere except at the Dirac points where it diverges.”

In the case of only an electric field, Eq. (5.8a) reduces to the anomalous velocity formula (3.6) derived before.

“As we can see, in the case of only an electric field, Eq. (5.8a) reduces to the anomalous velocity formula (3.6) we derived before.”

The latter provides another route for the Berry phase to enter transport problems in inhomogeneous situations, which can be caused by a non-uniform distribution function, or a spatially-dependent perturbation, or both.

“The latter provides another route for the Berry phase to enter the transport problems in inhomogeneous situations, which can be caused by a non-uniform distribution function, or a spatially-dependent perturbation, or both.”

The Fermi volume can be changed linearly by a magnetic field when the Berry curvature is nonzero.

“We show that the Fermi volume can be changed linearly by a magnetic field when the Berry curvature is nonzero.”

The constant density of states guaranteed by Liouville's theorem is no longer the case for Berry-phase modified dynamics.

“From a classical point of view, the constant density of states is guaranteed by the Liouvilles theorem, which states that the volume element is a conserved quantity during the time evolution of the system. 7 However, as we shall show below, this is no longer the case for the Berry-phase modified dynamics.”

The thermodynamic argument justifying the Streda relation is valid only for insulators because in metals the adiabaticity would break down.

“Note that this argument is valid only for insulators because in metals the adiabaticity would break down.”

The intrinsic contribution has been verified in CuCr2Se4-xBrx (Lee et al., 2004a,b), La1-xSrxCoO3 (Miyasato et al., 2007), Nd2Mo2O7 and Sm2Mo2O7 (Hanasaki et al., 2008), and Ga1-xMnx (Pu et al., 2008).

“The intrinsic contribution has been verified in CuCr 2 Se 4 -x Br x (Lee et al. , 2004a,b), La 1 -x Sr x CoO 3 (Miyasato et al. , 2007), Nd 2 Mo 2 O 7 and Sm 2 Mo 2 O 7 (Hanasaki et al. , 2008), Ga 1 -x Mn x (Pu et al. , 2008).”

• cites Lee et al., 2004a,bNot checked yet

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• cites Miyasato et al., 2007Not checked yet

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• cites Hanasaki et al., 2008Not checked yet

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The wave packet dynamics can be obtained from the time-dependent variational principles (Kramer and Saraceno, 1981).

“The wave packet dynamics can be obtained from the time-dependent variational principles (Kramer and Saraceno, 1981).”

• cites Kramer and Saraceno, 1981Not checked yet

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A scalar potential V(r) will induce a transverse velocity of the form Eq. (4.10).

“Therefore, in general a scalar potential V ( r ) will induce a transverse velocity of the following form”

Because the Berry curvature is generally k-dependent and the magnetic field is r-dependent, the phase-space volume ΔV changes during time evolution of the state variables (r,k).

“The fact that the Berry curvature is generally k -dependent and the magnetic fields is r -dependent implies that the phase-space volume ∆ V changes during time evolution of the state variables ( r , k ).”

The Lagrangian is given by Eq. (5.6), where ε_M(k) = ε_0(k) - B·m(k), with ε_0(k) being the unperturbed band energy.

“Using Eq. (4.5), we find the Langragian is given by, up to some unimportant total time-derivative terms (dropping the subscript c on r c and k c ),”

• Also stated in: § A. Equations of motion · p. 23 · plain

  “where ε M ( k ) = ε 0 ( k ) -B · m ( k ) with ε 0 ( k ) being the unperturbed band energy.”

Measurement of anomalous thermoelectric transport, such as the anomalous Nernst effect, can give further insight into the intrinsic mechanism of the anomalous Hall effect.

“Hence, the measurement of this type of transport, such as the anomalous Nernst effect, can give further insight of the intrinsic mechanism of the anomalous Hall effect.”

For band insulators with a nonzero integral of the Berry curvature (Chern number), discontinuous behavior of physical properties in a magnetic field is expected.

“Discontinuous behavior of physical properties in a magnetic field is therefore expected for band insulators with a nonzero integral of the Berry curvatures (Chern numbers).”

Streda (1982) derived a formula relating the Hall conductivity to the field derivative of the electron density at a fixed chemical potential.

“In the context of the quantum Hall effect, Streda (1982) derived a formula relating the Hall conductivity to the field derivative of the electron density at a fixed chemical potential”

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The velocity formula should stay the same whether the electric field enters through a time-dependent vector potential or through the gradient of the electrostatic potential, because it should be gauge invariant.

“In both cases, the velocity formula should stay the same because it should be gauge invariant.”

L11xy is proportional to the well-known anomalous Hall conductivity.

“This is proportional to the well-known anomalous Hall conductivity.”

• AssumptionThe authors state this as an assumption: We assume ∇µ = 0 and ∇T = 0.

The correct description of anomalous thermoelectric transport in ferromagnets requires knowledge of both the magnetic moment and orbital magnetization.

“It turns out the correct description of anomalous thermoelectric transport in ferromagnets requires the knowledge of both the magnetic moment and orbital magnetization.”

The time derivative of the spin dipole contributes to the total spin current (Culcer et al., 2004).

“The time derivative of the spin dipole contributes to the total spin current (Culcer et al. , 2004).”

• cites Culcer et al., 2004Not checked yet

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The paper recovered the usual expression for the cyclotron period, with the 2D electron density defined with the modified density of states.

“We thus recovered the usual expression for the cyclotron period, with the 2D electron density, Eq. (5.29), defined with the modified density of states.”

The dislocation situation is similar to the Aharonov-Bohm effect, with the dislocation playing the role of the solenoid and the Berry curvature Ω_rr the role of the magnetic field.

“What we have here is a situation similar to the Aharonov-Bohm effect (Aharonov and Bohm, 1959), with the dislocation playing the role of the solenoid, and the Berry curvature Ω rr the role of the magnetic field.”

• cites Aharonov and Bohm, 1959Not checked yet

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The agreement between calculated and numerical electron density confirms the Berry-phase modification to the density of states.

“which confirms the Berry-phase modification to the density of states.”

The electron density of states is modified by the Berry curvature.

“The electron density of states is also modified by the Berry curvature.”

If the Berry curvature vanishes, the density of states is constant.

“The density of states is a constant in this case.”

In the time-dependent case, the density of states D(r,q) has the same form as in the time-independent case.

“One can show that in the time-dependent case D ( r , q ) has the same form.”

The Berry curvature between r and q can be realized in deformed crystals as a quantity proportional to the strain and the electronic mass renormalization in the crystal (Sundaram and Niu, 1999).

“Finally, the Berry curvature between r and q can be realized in deformed crystals as a quantity proportional to the strain and the electronic mass renormalization in the crystal (Sundaram and Niu, 1999).”

• cites Sundaram and Niu, 1999Not checked yet

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Bird and Preston (1988) showed that this Berry phase can affect the electron diffraction pattern of a deformed crystal.

“Bird and Preston (1988) showed that this Berry phase can affect the electron diffraction pattern of a deformed crystal.”

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The formalism is applicable to dislocation strain fields, which are well defined except in a region of a few atomic spacings around the dislocation line.

“The above formalism is also applicable to dislocation strain fields, which are well defined except in a region of a few atomic spacings around the lin of dislocation.”

To first order in the spatial gradient, the density of states is given by Eq. (6.16).

“In many situations we are aiming at a first-order calculation in the spatial gradient. In this case, the density of states given by”

Due to the loss of tracking of λ, there is an uncertain quantum which is the second Chern number.

“Again, due to the loss of tracking of λ , there is an uncertain quantum which is the second Chern number.”

The adiabatic current induced by inhomogeneity is explicitly proportional to the spatial gradient.

“We can see that this current is explicitly proportional to the spatial gradient.”

The physics of quantum adiabatic transport and the quantum and anomalous Hall effect can be described from a unified point of view.

“The physics of quantum adiabatic transport and the quantum and anomalous Hall effect can be described from a unified point of view.”

In the low field limit (ω_c τ ≪ 1), the Berry phase induces a linear magnetoresistance.

“Next we consider the magnetotransport at the low field limit, i.e., ω c τ ≪ 1. In particular, we show that the Berry phase induce a linear magnetoresistance.”

The Berry curvature Ω_rt plays a role like the electric force.

“The Berry curvature Ω r t plays a role like the electric force.”

The magnetic-field-induced polarization implies an electromagnetic coupling of the form Eq. (6.35).

“such a magnetic-field-induced polarization implies that there is an electromagnetic coupling of the form”

If Ω(k) is smooth and W_kk' is localized, the relaxation time can be approximated by Eq. (5.40).

“If we assume Ω ( k ) is smooth and W kk ′ is localized, the relaxation time can be approximated by”

• AssumptionThe authors state this as an assumption: Ω(k) is smooth and W_kk' is localized.

When θ = π, the corresponding insulator is known as a 3D Z2 topological insulator.

“When θ = π , the corresponding insulator is known as a 3D Z 2 topological insualtor (Qi et al. , 2008).”

• cites Qi et al., 2008Not checked yet

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For a two-dimensional minimal model with h(r) having two components and H[λh(r)], the polarization can be written as Eq. (6.32).

“Consider a twodimensional minimal model with h ( r ) having two components. If we write H [ h ( r ); λ ] as H [ λ h ( r )], i.e., λ acts like a switch, the polarization can be written as”

A domain wall in the dimerization profile is known to carry fractional charge.

“Such a domain wall is known to carry fractional charge (Rice and Mele, 1982; Su et al. , 1979).”

• cites Rice and Mele, 1982Not checked yet

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• cites Su et al., 1979Not checked yet

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In the free electron limit (V→0), the deformation-induced gradient correction quantity vanishes.

“Note that in the free electron limit ( V → 0) this quantity vanishes.”

Non-canonical variables are a common feature of Berry-phase participated dynamics.

“The non-canonical variables is a common feature of Berry-phase participated dynamics (Littlejohn and Flynn, 1991).”

• cites Littlejohn and Flynn, 1991Not checked yet

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Xiao et al. (2007a) showed that the first-order contribution to polarization can be classified into a perturbative contribution due to wavefunction correction and a topological contribution from electron dynamics.

“Xiao et al. (2007a) showed that the first order contribution to the polarization can be classified into two categories: the perturbative contribution due to the correction to the wave function, and the topological contribution which is from the dynamics of the electrons.”

• cites Xiao et al. (2007a)Not checked yet

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If the perturbation is an electromagnetic field, the density of states equation (6.15) reduces to equation (5.13) under the substitution k = q + eA(r).

“If the perturbation is electromagnetic field, by the variable substitution k = q + e A ( r ), Eq. (6.15) reduces to Eq. (5.13).”

If the order parameter has period l_y in the y-direction, the polarization quantum in the x-direction is given by Eq. (6.31).

“If we assume the order parameter has period l y in the y -direction, the polarization quantum in the x -direction is given by”

• AssumptionThe authors state this as an assumption: The order parameter has period l_y in the y-direction.

Essin et al. (2008) considered the polarization induced by a magnetic field, which can be regarded as arising from a spatial inhomogeneity in the vector potential A(r).

“Using the above formalism, Essin et al. (2008) considered the polarization induced by a magnetic field, which can be regarded as arising from a spatial inhomogeneity in the vector potential A ( r ).”

• cites Essin et al. (2008)Not checked yet

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Under a weak uniform electric field E, according to the semiclassical equations of motion, the quasi-momentum of an electron wavepacket is given by Eq. (7.4).

“Under a weak uniform electric field E , according to the semiclassical equations of motion, the quasi-momentum of an electron wavepacket is simply (see Eq. (5.8))”

Before applying a magnetic field, the energy dispersion near the Dirac point in graphene is linear, E(k) = hbar v_F k.

“Before applying a magnetic field, the energy dispersion near the Dirac point is linear, E ( k ) = ¯ hv F k .”

The center-of-mass variables r_c and k_c in the semiclassical dynamics in Eq. (5.8) are not canonical variables since their Poisson brackets are not of the canonical form.

“For example, the center-ofmass variables r c and k c in the semiclassical dynamics in Eq. (5.8) are not canonical variables since their Poisson brackets are not of the canonical form (Duval et al. , 2006a; Xiao et al. , 2005),”

• cites Duval et al., 2006aNot checked yet

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• cites Xiao et al., 2005Not checked yet

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For a simple one-dimensional lattice with inversion symmetry, the Berry phase Γ_Cm can only have two values, 0 or π.

“For a simple one-dimensional lattice with inversion symmetry, the Berry phase Γ C m can only have two values, 0 or π (Zak, 1989b), as discussed earlier in Sec. II.C.”

• cites Zak, 1989bNot checked yet

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Due to the Berry phase, the average positions are displaced from the positive ions located at am.

“Due to the Berry phase, they are displaced from the positive ions located at am .”

The average positions 〈r_c〉_m are the average positions of the Wannier function.

“Such average positions 〈 r c 〉 m are the average positions of the Wannier function (Vanderbilt and King-Smith, 1993).”

• cites Vanderbilt and King-Smith, 1993Not checked yet

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For a moving domain wall in a thin magnetic wire, the Berry curvature Ω_rt will induce an electromotive force, resulting in a voltage difference between the two ends.

“On the other hand, for a moving domain wall in a thin magnetic wire, the Berry curvature Ω rt will induce an electromotive force, which results in a voltage difference between the two ends.”

The Bohr-Sommerfeld quantization method may give inconsistent quantization rules because choices of conjugate variables are not unique.

“However, since the choices of conjugate variables are not unique, the Bohr-Sommerfeld quantization method may give inconsistent quantization rules.”

The problem can be fixed by using integrals over closed loops in phase space with integer m_i and Maslov indices ν_i (EBK quantization).

“This problem can be fixed if, instead of Eq. (7.1), one uses the following integrals,”

• cites Tabor, 1989Not checked yet

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If the energy dispersion near a degenerate point is linear, then the cyclotron orbit will acquire a Berry phase Γ_C = π, independent of the shape of the orbit.

“It is known that, if the energy dispersion near a degenerate point is linear, then the cyclotron orbit will acquire a Berry phase Γ C = π , independent of the shape of the orbit (Blount, 1962b).”

• cites Blount, 1962bNot checked yet

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According to Eqs. (6.9), this Berry curvature field acts on the electrons as an effective magnetic field.

“According to Eqs. (6.9), this field acts on the electrons as an effective magnetic field.”

The experimental observation of a quantum Hall plateau at zero energy is a direct consequence of the Berry phase.

“The experimental observation of a quantum Hall plateau at zero energy is thus a direct consequence of the Berry phase (Novoselov et al. , 2005, 2006; Zhang et al. , 2005).”

• cites Novoselov et al., 2005, 2006Not checked yet

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• cites Zhang et al., 2005Not checked yet

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Bloch oscillation is not commonly observed in laboratory because in a usual solid the electron scattering time is shorter than the period T_B by several orders of magnitude, destroying phase coherence within a tiny fraction of a period.

“Second, the fascinating phenomenon of Bloch oscillation is not commonly observed in laboratory for the following reason: In an usual solid, the electron scattering time is shorter than the period T B by several orders of magnitude. Therefore, the phase coherence of the electron is destroyed within a tiny fraction of a period.”

This Berry curvature induced emf has been experimentally measured recently.

“This Berry curvature induced emf has has been experimentally measured recently (Yang et al. , 2009).”

• cites Yang et al., 2009Not checked yet

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The orbit in k-space can be obtained by re-scaling the r-space orbit in Eq. (7.9) with a linear factor of λ_B^2, followed by a rotation of ninety degrees, where λ_B = sqrt(hbar/eB).

“The orbit in k -space can be obtained by re-scaling the r -space orbit in Eq. (7.9) with a linear factor of λ 2 B , followed by a rotation of ninety degrees, where λ B ≡ √ ¯ h/eB is the magnetic length (Ashcroft and Mermin, 1976).”

• cites Ashcroft and Mermin, 1976Not checked yet

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With the semiclassical equation, the Bohr-Sommerfeld condition in Eq. (7.3) can be written as Eq. (7.9) with ν=2.

“the Bohr-Sommerfeld condition in Eq. (7.3) can be written as (notice that ¯ h q c = ¯ h k c -e A , and ν = 2),”

In the wavepacket formulation of Bloch electrons, both r_c and q_c are treated as generalized coordinates, and their conjugate momenta are hbar q_c and hbar A respectively.

“In the wavepacket formulation of Bloch electrons, both r c and q c are treated as generalized coordinates. With the Lagrangian in Eq. (5.7), one can find their conjugate momenta ∂L/∂ ˙ r c and ∂L/∂ ˙ q c , which are equal to ¯ h q c and ¯ h 〈 u | i∂u/∂ q c 〉 = ¯ h A respectively (Sundaram and Niu, 1999).”

• cites Sundaram and Niu, 1999Not checked yet

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For a ferromagnetic thin film described by Hamiltonian Eq. (6.36), the simple momentum-dependence dictates that all k-dependent Berry curvatures vanish.

“The simple momentum-dependence of the Hamiltonian dictates that all k -dependent Berry curvatures vanish.”

An orbit C in k-space lies on the intersection of a plane perpendicular to the magnetic field and the constant-energy surface.

“From Eq. (5.25b), it is not difficult to see that an orbit C in k -space lies on the intersection of a plane perpendicular to the magnetic field and the constant-energy surface (Ashcroft and Mermin, 1976).”

• cites Ashcroft and Mermin, 1976Not checked yet

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Knowledge of the Bloch energy, the Berry curvature, and the magnetic moment in the semiclassical theory constitutes sufficient information for building the re-quantized theory.

“It will become clear that, the knowledge of the Bloch energy, the Berry curvature, and the magnetic moment in the semiclassical theory constitute sufficient information for building the re-quantized theory.”

The value of the Maslov index is not always evident in the Bohr-Sommerfeld method.

“First, the value of the Maslov index is not always evident.”

The Berry curvature Ω_rr will drive a Hall current, just like the ordinary Hall effect.

“The Berry curvature Ω rr will drive a Hall current, just like the ordinary Hall effect (Bruno et al. , 2004; Ye et al. , 1999).”

• cites Bruno et al., 2004Not checked yet

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• cites Ye et al., 1999Not checked yet

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When a uniform B field is applied to a solid, the electron executes a cyclotron motion in both r-space and k-space.

“When an uniform B fi eld is applied to a solid, the electron would execute a cyclotron motion in both the r -space and the k -space.”

In magnetic materials with domain walls or spin textures, Berry curvatures originate only from the nontrivial real space configuration of the system.

“In this subsection we discuss the Berry curvatures which originate only from the nontrivial real space configuration of the system.”

Ω_rt acts on the electrons as an effective electric field.

“Similarly, Ω r t acts on the electrons as an effective electric field.”

The hyperorbit motion can be quantized using the Bohr-Sommerfeld quantization rule.

“Similar to the cyclotron orbit, the hyperorbit motion can also be quantized using the Bohr-Sommerfeld quantization rule (see Eq. (7.3)).”

This rule can be derived with the help of the magnetic translation symmetry.

“This rule can be derived with the help of the magnetic translation symmetry (Dana et al. , 1985).”

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The states on opposite sides of the MBZ boundaries represent the same physical state.

“The states on opposite sides of the MBZ boundaries represent the same physical state.”

Therefore, states on opposite sides of the MBZ boundaries can only differ by a phase factor.

“Therefore, they can only differ by a phase factor.”

Therefore, there is no cyclotron motion of k even though there is a magnetic field B0.

“Therefore, there is no cyclotron motion of k (even though there is a magnetic field B 0 ).”

The semiclassical equations of motion valid under the one-band approximation have exactly the same form as Eq. (5.8), with hbar k, E0(k), and B reinterpreted as the magnetic momentum, magnetic band energy, and extra magnetic field δB respectively.

“The semiclassical equations of motion that is valid under the one-band approximation have exactly the same form as Eq. (5.8). One simply needs to reinterpret ¯ h k , E 0 ( k ), and B in Eq. (5.8) as the magnetic momentum, the magnetic band energy, and the extra magnetic field δ B respectively (Chang and Niu, 1995, 1996).”

• cites Chang and Niu, 1995, 1996Not checked yet

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According to Laughlin's gauge-invariance argument, each isolated magnetic Bloch band carries a quantized Hall conductivity.

“According to Laughlin's gauge-invariance argument (Laughlin, 1981), each of the isolated magnetic Bloch band carries a quantized Hall conductivity (see Sec. II.B and Sec. III.C).”

• cites Laughlin, 1981Not checked yet

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The extra vector potential ∇f can be removed by a gauge transformation Eq. (8.4).

“The extra vector potential ∇ f can be removed by a gauge transformation,”

The same cannot be true for the k2 boundaries, otherwise the topology will be too trivial to accommodate the quantum Hall conductivity.

“The same cannot also be true for the k 2 boundaries, otherwise the topology will be too trivial to accommodate the quantum Hall conductivity.”

The phase has to be non-integrable in order to account for the Hall conductivity.

“For one thing, the phase has to be non-integrable in order to account for the Hall conductivity.”

The Bloch energy, Berry curvature, and orbital moment of the semiclassical theory contain sufficient information for building a quantum theory that accounts for all physical effects to first order in external fields.

“Second, the Bloch energy, Berry curvature, and orbital moment of the semiclassical theory contains sufficient information for building a quantum theory that accounts for all physical effects to first order in external fields.”

This equation determines the Hall conductivity (mod q) of a magnetic Bloch band.

“This equation determines the Hall conductivity (mod q ) of a MBB (Dana et al. , 1985).”

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The quantized Hall conductivity is closely related to the nontrivial topological property of the magnetic Bloch state.

“This is closely related to the nontrivial topological property of the magnetic Bloch state (Kohmoto, 1985; Morandi, 1988).”

• cites Kohmoto, 1985Not checked yet

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• cites Morandi, 1988Not checked yet

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The distribution of Hall conductivities among the split subbands follows a simple rule first discovered by Thouless et al. (1982).

“Furthermore, the distribution of Hall conductivities among the split subbands follows a very simple rule first discovered by Thouless et al. (1982) et al .”

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When the lattice potential is much stronger than the magnetic field, if each plaquette encloses a magnetic flux (p/q)φ0, then each Bloch band splits to q subbands.

“It was found that, if each plaquette encloses a magnetic flux ( p/q ) φ 0 , then each Bloch band would split to q subbands (Hatsugai and Kohmoto, 1990; Kohmoto, 1989).”

• cites Hatsugai and Kohmoto, 1990Not checked yet

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• cites Kohmoto, 1989Not checked yet

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When both B and Ω are nonzero, applying both Peierls substitutions simultaneously is not enough to produce the correct commutation relations, mainly because of the non-trivial factor κ.

“When both B and Ω are nonzero, applying both of the Peierls substitutions simultaneously is not enough to produce the correct commutation relations, mainly because of the non-trivial factor κ there.”

Because of the vector potential, the Hamiltonian H no longer has the lattice translation symmetry.

“Because of the vector potential, the Hamiltonian H above no longer has the lattice translation symmetry.”

Since the lattice symmetry of the charge density is not broken by a uniform magnetic field, one should be able to define translation operators that differ from the usual ones only by phase factors.

“However, since the lattice symmetry of the charge density is not broken by an uniform magnetic field, one should be able to define translation operators that differ from the usual ones only by phase factors (Lifshitz and Landau, 1980).”

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In order for the integral (1/2π)∮∂MBZ dk·A(k) to be the Hall conductivity σ_H (in units of h/e^2), the integer in Eq. (8.16) has to be equal to σ_H.

“In order for the integral (1 / 2 π ) ∮ ∂ MBZ d k · A ( k ) (which is nonzero only along the upper horizontal boundary) to be the Hall conductivity σ H (in units of h/e 2 ), the integer in Eq. (8.16) obviously has to be equal to σ H .”

The operator T_a defined as the magnetic translated state has the property [H,T_a]=0.

“The operator T a being defined this way has the desired property that [ H,T a ]=0.”

The non-commutativity of magnetic translations is a result of the Aharonov-Bohm phase.

“That is, the non-commutativity is a result of the Aharonov-Bohm phase.”

If the magnetic field is much stronger than the lattice potential, and each plaquette has a magnetic flux φ=(p/q)φ0, then after turning on the lattice potential each Landau level splits to p subbands.

“In this case, if each plaquette has a magnetic flux φ = ( p/q ) φ 0 , then after turning on the lattice potential, each LL will split to p subbands.”

The term in the spin dynamics proportional to the Berry connection is in fact the spin-orbit coupling.

“Later we will see that it is in fact the spin-orbit coupling.”

The semiclassical electron dynamics with an Abelian Berry curvature can be extended to cases where the energy bands are degenerate or nearly degenerate (e.g., due to spin).

“Such a formalism can be extended to the cases where the energy bands are degenerate or nearly degenerate (eg., due to spin) (Culcer et al. , 2005; Shindou and Imura, 2005).”

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• cites Shindou and Imura, 2005Not checked yet

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In addition to the Berry curvature, impurity scattering is another source of the extrinsic SHE.

“In addition to the Berry curvature, impurity scattering is another source of the (extrinsic) SHE.”

For the non-Abelian case, the Berry curvature F is often proportional to the spin S.

“For the non-Abelian case, the Berry curvature F is often proportional to the spin S (see Secs. VIII.C and IX.E).”

Both the distribution of the σ_r's and the pattern of splitting can be determined entirely within the semiclassical formulation.

“Therefore, both the distribution of the σ r 's and the pattern of splitting can be determined entirely within the semiclassical formulation.”

Because of the spinor degree of freedom, the Berry curvature and magnetic moment become vector-valued matrices instead of usual vectors.

“However, because of the spinor degree of freedom, the latter two become vector-valued matrices, instead of the usual vectors.”

This is the source of the intrinsic spin Hall effect (SHE) in semiconductors predicted by Murakami et al. (2003).

“This is the source of the intrinsic spin Hall effect (SHE) in semiconductors predicted by Murakami et al. (2003).”

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Intrinsic SHE has also been theoretically predicted in metals.

“Indeed, intrinsic SHE has also been theoretically predicted in metals (Guo et al. , 2008).”

• cites Guo et al., 2008Not checked yet

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Because the MBZ is q-fold degenerate, the number of energy levels produced by these hyperorbits is given by Eq. (8.26).

“Because the MBZ is q -fold degenerate (see Sec. VIII.A), the number of energy levels produced by these hyperorbits are (Chang and Niu, 1995)”

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Since the total number of hyperorbits in the parent band can be determined by the quantization rule, one can pin down the value of σ_open_r with the help of the sum rule σ_parent_H = ∑_r σ_r.

“However, since the total number of hyperorbits in the parent band can be determined by the quantization rule, we can easily pin down the value of σ open r with the help of the sum rule: σ parent H = ∑ r σ r .”

The trajectories of spin-up and spin-down electrons are parted toward opposite directions transverse to the electric field.

“That is, the trajectories of spin-up and spin-down electrons are parted toward opposite directions transverse to the electric field.”

This is the same skew scattering (or Mott scattering) in AHE.

“This is the same skew scattering (or Mott scattering) in AHE (see Sec. III.D.1).”

Because degenerate Bloch states have multiple components, the Berry curvature becomes a matrix with non-Abelian gauge structure.

“Because the degenerate Bloch states have multiple components, the Berry curvature becomes a matrix with non-Abelian gauge structure.”

If the wavepacket is narrowly distributed around q_c, it is possible to write the magnetic moment as the spinor average of M.

“If the wavepacket is narrowly distributed around q c , then it is possible to write it as the spinor average of the following quantity (Culcer et al. , 2005),”

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If tunnelling between degenerate hyperorbits is taken into account, each energy level will be broadened into an energy band.

“If one further takes the tunnelling between degenerate hyperorbits into account (Wilkinson, 1984a), then each energy level will be broadened into an energy band.”

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As a result, the frequency of the hyperorbit motion can be very low.

“As a result, the frequency of the hyperorbit motion can be very low.”

It is possible to detect the hyperorbit using resonant absorption of ultrasonic wave or conductance oscillation in an electron focusing device.

“Nevertheless, it is possible to detect the hyperorbit using, for example, resonant absorption of ultrasonic wave or the conductance oscillation in an electron focusing device.”

There can be a net transverse current if the populations of spin-up and spin-down electrons are different, such as in a ferromagnet.

“There can be a net transverse current if the populations of spin-up and spin-down electrons are different, such as the case in a ferromagnet.”

According to Laughlin's argument, each isolated subband should have its own integer Hall conductivity.

“According to Laughlin's argument, each of the isolated subband should have its own integer Hall conductivity.”

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The spin Hall current can still be nonzero.

“However, the spin Hall current can still be nonzero.”

R and F have the same structure as the gauge potential and gauge field in the non-Abelian SU(2) gauge theory.

“Here, R and F also have the same structure as the gauge potential and gauge field in the non-Abelian SU (2) gauge theory (Wilczek and Zee, 1984).”

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This leads to the anomalous Hall effect.

“This then leads to the anomalous Hall effect.”

Such a projection of the Hilbert space almost always generates a Berry curvature.

“As we shall see in Sec. IX.E, such a projection of the Hilbert space almost always generates a Berry curvature.”

Therefore, the SHE should be common in semiconductors or other materials.

“Therefore, the SHE should be common in semiconductors or other materials.”

Because of its open trajectory, the above analysis fails for the nesting orbit since the first term in Eq. (8.27) also contributes to the Hall conductivity.

“Because of its open trajectory, the above analysis fails for the nesting orbit since the first term in Eq. (8.27) also contributes to the Hall conductivity.”

As a result of band splitting, the integer Hall conductivity σ_H of the parent band is split to a distribution of integers σ_r (there are q of them).

“That is, as a result of band splitting, the integer Hall conductivity σ H of the parent band is split to a distribution of integers σ r (there are q of them).”

If F is proportional to S, then in the presence of an electric field, the anomalous velocity is proportional to E × S.

“If this is true, then in the presence of an electric field, the anomalous velocity is proportional to E × S .”

If the populations of different spins are equal, the net electric Hall current is zero.

“If the populations of different spins are equal, then the net electric Hall current is zero.”

The projection from the full space with eight bands to the HH-LH complex of four bands generates a Berry curvature F_p = -(2 V^2 / 3 E_g^2) J, where J is the spin-3/2 matrix.

“The projection from the full space with eight bands to the HH-LH complex of four bands generates a Berry curvature F p = -(2 V 2 / 3 E 2 g ) J (Chang and Niu, 2008), where J is the spin-3/2 matrix.”

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Because of the huge gap between branches, interbranch tunnelling happens (and the semiclassical theory fails) only if the field is so strong that electron-positron pair-production can no longer be ignored.

“Furthermore, because of the huge gap between branches, interbranch tunnelling happens (and the semiclassical theory fails) only if the field is so strong that electron-positron pair-production can no longer be ignored.”

In this case, even the lattice potential in a solid can be considered as a semiclassical perturbation.

“In this case, even the lattice potential in a solid can be considered as a semiclassical perturbation.”

The canonical variables r and p accurate to linear order of the fields are related to the center-of-mass variables as given by Eq. (9.11).

“The canonical variables r and p accurate to linear order of the fields are related to the center-of-mass variables as follows (Chang and Niu, 2008),”

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The dipole term e E · R yields a spin-orbit coupling whose coefficient α and form are the same as the Rashba coupling.

“The coefficient α and the form of the spin-orbit coupling are the same as the Rashba coupling (Bychkov and Rashba, 1984; Rashba, 1960).”

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For both the Dirac electron and the semiconductor electron, the dipole term is closely related to the spin-orbit coupling.

“Although the exact form of the Berry connection R depends on the physical model, we will show that for both the Dirac electron (IX.D) and the semiconductor electron (IX.E), the dipole term is closely related to the spin-orbit coupling.”

The wavepacket in the conduction band spontaneously rotates with respect to its own center of mass.

“The wavepacket in the conduction band also spontaneously rotates with respect to its own center of mass.”

The effective Hamiltonian alone is not enough for correct prediction if the physical variables have not been identified properly.

“The effective Hamiltonian itself is not enough for correct prediction, if the physical variables have not been identified properly.”

If one chooses the eight bands as the full space, the Berry curvatures of the heavy hole and the light hole get new contributions from P F_p P.

“Instead of the HH-LH complex, if one chooses the eight bands in Fig. 15 as the full space, then the Berry curvatures of the heavy hole and the light hole will get new contributions from P F p P .”

Starting from theory II instead of theory I as the parent theory would introduce errors P F_p P and P M_p P.

“This means that starting from theory II, instead of theory I, as the parent theory, one would have the errors P F p P and P M p P .”

Therefore, the effective quantum theory can be smooth globally.

“Therefore, the effective quantum theory can be smooth globally.”

The Pauli Hamiltonian can be obtained from blockdiagonalizing the Dirac Hamiltonian using a unitary transformation.

“The Pauli Hamiltonian can also be obtained from blockdiagonalizing the Dirac Hamiltonian using an unitary transformation.”

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The discrepancy between the two transverse velocities can be traced to the difference between the two position variables r_c and r.

“The source of this discrepancy can be traced back to the difference between the two position variables: r c and r (see Eq. (9.11)).”

The two branches are separated by a huge energy gap mc^2.

“This two branches are separated by a huge energy gap mc 2 .”

The Berry curvature and magnetic moment for theory III when starting from the parent space (F′ and M′) differ from F and M because they are obtained from summations with more outside states from the full space.

“They are different from F and M since the former are obtained from the summations with more outside states from the full space.”

If one starts from the parent space, the Berry curvature and the magnetic moment for theory II are zero.

“On the other hand, if one starts from the parent space, then the Berry curvature and the magnetic moment for theory II is zero (instead of F p and M p ).”

The equation for ̇r_c should be regarded as the correct one because it is based on the Dirac theory.

“One should regard the equation for ˙ r c as the correct one since it is based on the Dirac theory (also see Bliokh (2005)).”

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