On quantum Hall effect, Kosterlitz-Thouless phase transition, Dirac magnetic monopole, and Bohr-Sommerfeld quantization

We discuss how the Bohr-Sommerfeld quantization condition permeates the relationships between the quantization of Hall effect, the Berezenskii-Kosterlitz-Thouless vortex quantization, the Dirac magnetic monopole, the Haldane phase, the contact resistance in closed mesoscopic circuits of quantum physics. This paper is motivated by the recent derivation by one of the authors of the topological Chern number of the integer quantum Hall effect in electrical conductivity using a novel phase-space nonequilibirum quantum transport approach. The topological invariant in (~p, ~q; E, t)-phase space occurs to first-order in the gradient expansion of the nonequilibrium quantum transport equation. The Berry curvature related to orbital magnetic moment is also calculated leading to the quantization of orbital motion and edge states for 2-D systems. All of the above physical phenomena maybe unified simply from the geometric point of view of the old Bohr-Sommerfeld quantization, as a theory of Berry connection or a U (1) gauge theory


Introduction
There are two general themes that will be addressed in this paper, namely, transport and vortex/precessionmotion quantization of low-dimensional systems, and stationary quantization of confined motion in phase space due to oscillatory dynamics or compactification of space and time (i.e., particle in a box, Brillouin zone, and Matsubara time zone or Matsubara quantized frequencies) for steady-state system. The former is a fairly recent phenomenon which has become the springboard of modern condensed-matter physics research, whereas the latter is as old as the beginning of quantum mechanics of atomic systems. Although, the latter is old it continues to be interesting and a powerful way of getting a handle of complex problems, and indeed it has now merited revisits since its innocent looking formalism may actually be of rigorous geometric origin. This is based on the geometrical concept of connection [1]. Our task is to show that the old is the forebear of the new and permeates or pervades the whole of modern topological quantum physics. The geometrical concept of connection also allows us to deduce the quantized Dirac magnetic monopole without the use of Dirac semi-infinite string [2].
The quantization of Hall conductance of electrical conductivity in a two-dimensional periodic potential was first explained by Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) [3] using the Kubo currentcurrent correlation. A similar approach was employed by Streda [4]. Earlier, Laughlin [5], and later Halperin [6], study the effects produced by changes in the vector potential on the states at the edges of a finite system, where quantization of the conductance is made explicit, but how their result is insensitive to boundary conditions was not clear. In contrast, the use of Kubo formula by TKNN is for bulk two-dimensional conductors.
These theoretical works were motivated by the 1980 Nobel Prize winning experimental discovery of von Klitzing, Dorda, and Pepper [7] on the quantization of the Hall conductance of a two-dimensional electron gas in a strong magnetic field. The strong magnetic field basically provides the gapped energy structure for the experiments. In the TKNN approach, periodic potential in crystalline solid is being treated. A strong magnetic field is not needed to provide the gapped energy structure in their theory, only peculiar gapped energy-band structures. In principle, in the presence of electric field the discrete Landau levels is replaced by the unstable discrete Stark ladder-energy levels [8,9] due to compactification of space into a torus, known as the Born-von Karman boundary condition.
Our new approach to integer quantum Hall effect (IQHE) makes use of the real-time superfield and lattice Weyl transform nonequilibirum Green's function (SFLWT-NEGF) [10] quantum transport equation to firstorder in the gradient expansion [11,12], the topological Chern number of the integer quantum Hall effect (IQHE) for two-dimensional systems is obtained. This new transport approach avoids the conventional use of Kubo formula originally employed by TKNN [3]. The topological invariant of quantum transport in ( p, q; E, t)-space is thus identified. The Kubo current-current formula strictly follows from real-time nonequilibrium quantum superfield theory of transport physics adapted to time-dependent electric fields. The orbital magnetic moment and its related Berry curvature is also calculated leading to quantized orbital motion and edge states.
Another Nobel prize winning theoretical discovery [3], and confirmed experimentally, is the so-called Kosterlitz-Thouless phase transition (KTPT) first discovered in X-Y model of spin systems. This goes well-beyond the well-established theorem, the so-called Mermin-Wagner theorem [13], or Mermin-Wagner-Berezinskii theorem, which states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions d ≤ 2. The vortex and antivortex solutions of KTPT goes beyond Ginzburg-Landau symmtery breaking phase transition which been quite successful. On the other hand, the vortex solutions of KTPT has a topological content (basically topolgical defects) and bears resemblance to the B-S quantization condition. The B-S quantization also crept into the theory of another new topological phase of matter, so called Haldane phase of odd-integer spin chain.
The recent developments in the cross-fertilization of pure mathematics and physics is invigorated due to recent discoveries in physics which have relevance to pure mathematics. The IQHE and fractional FQHE have recently been geometrically formalized and unified based on Hecke operators and Hecke eigensheaves (mathematical replacement of the term eigenfunctions) i.e., of the geometric Langlands program by Ikeda [14]. In particular the plateaus of the Hall conductance are associated by the Hecke eigensheaves. Moroever, the fractal energy spectrum of a tight-binding Hamiltonian in a magnetic field, known as the Hofstadter's butterfly, as been associated with Langlands duality in the quantum groups. Indeed, the Langlands program in pure mathematics may carry several realization in theoretical physics.
As it turns out, the old Bohr-Sommerfeld quantization rules has recently been realized to have much deeper ramifications in quantum physics. In a much simpler terms in some cases, it has to do with the counting of discrete " -voxcells" of actions in phase-space of generalized canonical variables measured in units of Planck's constant, h, or number of quantum flux in the case of magnetic fields. By counting operation means that the results belongs to the domain of integers, Z. Here we show that the old quantization scheme have bearings on Landau-level degeneracy, massive Dirac magnetic monopole, quantzation of orbital magnetic moment and angular momentum, and Bohr-Sommerfeld quantization in superfluidity, superconductivity, and quantum field theory.

Geometric origin of the Bohr-Sommerfeld quantization
We give here a geometrical point of view of the Bohr-Sommerfeld (B-S) quantization rule given by, 1 2π The energy spectrum of hydrogen atom agreed exactly with observed spectrum as obtained by Sommerfeld. The B-S theory was applied with varying success to other systems. In fact, it is remarkable that the energy spectrum of Dirac equation for an electron in an atom obtained by Sommerfeld also agrees exactly [15]. With a minor modifications B-S quantization gives the energy spectrum of π meson, which can be obtained by solvimg the Klein-Gordon equation [16].
Here we give a much simpler and intuitive geometric derivation of the B-S quantization condition, as a contour integral of the gradient of geometric phase or Berry connection. Let |n, q be the basis eigenvector of the eigenstates of the Schrödinger equation, parametrized by q, H (q) |n, q = E n |n, q Note from the quantum mechanical viewpoint, p i and dq i in B-S quantization are c-numbers not operators. These c-numbers must actually be derived from the Schrödinger wavefunctions, adiabatically, i.e., as diagonal matrix element of the momentum operator, i ∇ q when operating on the eigenvector |n, q 1 2π where we made use of the result of the change of phase in parallel transporting Schrödinger wavefunction in q-space, where n =⇒ α denotes the discrete energy level 1 . Furthermore, where we defined A as acting like a vector potential. In terms of the introduced 'vector potential', A, the B-S quantization condition now reads Therefore we arrive at the Bohr-Sommerfeld quantization rules for closed orbits purely from a geometric considerations, where n, q| ∇ qi |n, q ≡: e c A is the Berry connection. It does appear that the B-S quantization rules have an exact geometric origin, as well as confirming the Planck's discretization of phase space in units of Planck's constant h. Therefore, it is expected that B-S condition will play an important role in topological band theory, as will be shown here. In what follows we give various examples where the B-S geometric quantization conditon enters in the analyses.

The B-S condition and quantization of angular momentum
An immediate case of a 'closed orbits' deal with either orbital or spin angular momentum. The Schrödinger equation for angular momentum about the z-axis is given by whereĴ z is the angular momentum operator for the z-component. Therefore, we obtain the B-S condition as, Therefore the eigenvalues of observable component, quantized in units of Planck action. This B-S quntization condition has been used by Haldane [18] in his theory of a new state of matter now known as the Haldane phase.

The B-S phase factor and Schrödinger equation
The Berry phase and Berry curvature was originally derived by Berry [1] using the Schrödinger equation whose Hamiltonian depends on a time-dependent parameter, R (t). This specifically made clear in the case of the Born-Oppenheimer approximation in molecular physics, where the Berry connection exactly acts like a vector potential in the effective Schrödinger equation for the slow variable [10]. Therefore the wavefunction evolves both in time and parameter space. However, because of the presence of the dynamical phase factor in the wavefunction, the B-S phase factor is not the only phase factor and hence the B-S quantization condition, on grounds of wavefunction uniqueness, cannot be implemented. This is left as integral around a contour in parameter R-space, under time compactification (i.e., 'time-Brillouin zone' or 'temporal box': t = 0 to T ) of steady-state condition. We have where the B-S phase or geometrical phase factor is given by The B-S condition and Landau-level degeneracy Indeed, the quantization condition in terms of the vector potential, Eq. (1) materializes in the method of counting of Planck states in phase space for the Landau circular orbits. This is the calculation of the Landau-level (L-L) degeneracy. Classically this may be approximated by where r is the classical radius of Landau orbits in a uniform magnetic fields and the system area is given by A = πR 2 . Equation (3) where Φ is the total magnetic flux, and φ o is the quantum flux. This is a realization of the B-S quantization condition given in Eq. (1). This quantized orbital motion leads to edge states and integer quantum Hall effect under uniform magnetic fields. The general analysis of edge states marks the works of Laughlin [5], and Halperin [6].

Peierls phase factor and Wilson loops
The above argument on confined or bounded motion can be made precise by noting that localized wavefunctions in energy-band theory with vector potential, either real electromagnetic or Berry connection, always carry the so-called Peierls phase factor. This is well-known since the early days of solid-state physics. Thus, 'bringing' (i.e., using magnetic translation operator defined below) a localized wavefunction around a closed loop (in modern terminology the so-called Wilson loop or plaquett) would acquire a total phase factor determined by Eq. (1), by virtue of uniqueness. This means that for electromagnetic vector potential, we have, which again follows from the Bohr-Sommerfeld quantization condition. We note that the Landau orbits are confined in phase space and the amount of "h-voxcells" or more appropriately the number of h-pixcels of phase space enclosed by the orbits in units of Planck's constant is quantized, i.e., Z. We can even say that any oscillatory and harmonic motion in phase space entails some interdependence of canonical coordinates so as to enclose an integral number of Planck's states, i.e., h-pixcels. In the case of magnetic fields, this translate to the number of quantum fluxes. The minimal coupling of canonical momentum under a uniform magnetic field provides the natural interdependence of the canonical conjugate variables, where the dependence on the coordinates is only provided by the vector potential.

Peierls phase factor and Dirac magnetic monopole
The classical Maxwell's equations are If Dirac magnetic monopole exists, giving a magnetic charge density, ρ m , then we have a complete duality of electric and magnetic fields, The electric-magnetic duality is characterized by the following replacements, The four equations in Eq. (5) is condensed into two using complex fields, which manifest additional symmetry of the electric-magnetic duality. Although, classical arguments persist against the existence of magnetic monopole, quantum theory is more favorable of its existence, as will be shown below.
Here, we can continue to make use of the Peierl's phase-factor arguments to deduce the existence of Dirac magnetic monopole, instead of the use of infinitely-long/thin Dirac string originally employed by Dirac [2]. This done by creating a tetrahedron or bounded 3-D domain using the magnetic translation operator to define the surfaces of the bounded region. The 'pointed tip of the pen' is served by a localized function represented by the magnetic Wannier function translated between lattice sites to define a tetrahedron.
For our purose, it is convenient to formalize the magnetic translation operator in the presence of vector potential defined by where A ( r) is the vector potential, choosen in symmetric gauge, A ( r) = 1 2 B × r, and q is the crystal lattice vector. 2 T ( q) generates all the magnetic Wannier functions belonging to a band index λ from a given Wannier function centered at the origin. Therefore, operating on the lattice-position eigenvector centered at the origin, we have Using the Baker-Campbell-Hausdorff (BCH) formula, we have This means that the phase of Eq. (10) is either zero or 2π× integer, as expected since application of the resulting total magnetic translation, T (− q 1 ) T (− q 2 ) T (− q 3 ) to a localized Wannier function, w ( r, o), should be equivalent to where q ≡ ( q 1 + q 2 + q 3 ) in the last line consistent with Eq. (6). Here a i =⇒ q i in the text.
The total phase of Eq. (9) suggest the following integrals where d s denotes the incremental surface in the r-coordinates (not the lattice coordinates). For the solution ∇ · B = 0 then Φ = 0. However, there is another solution for the phase, which can be written in detail as which clearly exhibit an expression with all the magnetic fluxes emerging from the tetrahedron. Without affecting the validity of Eq. (11), the total phase of Eq. (9) for all emerging magnetic flux of Eq. (12) must satisfy the following expression In other words if there is a magnetic charge e m , '+' for monopoles and '−' for antimonopoles. The magnetic field B is such that then it follows that the magnetic charge, e m , is quantized, which yields exactly the quantized Dirac magnetic monopole, originally given by Dirac employing a thought experiment of semi-infinite thin magnetic string, where c 2e = e 2α = 137 2 e (α is the fine structure constant = e 2 c ) is the unit magnetic charge. Now therefore, would be physically consistent if ∇ · B = 0 outside a singularity and ∇ · B = e m only at the core. This allows us to propose that where δV core is an infinitisimal domain containing the singularity at r = 0 or the quantum flux is emerging through a spherical surface surrounding the point of singularity. Clearly a magnetic monopole is infinitely localized and therefore must be hugely massive. Thus it requires very high-energy not currently readily available in experimental physics to detect it. Moreover, it may always occurs in pairs, namely, monopole and anti-monopole pair (i.e., north and south pole, similar to vortex anti-vortex pairs of the X-Y model of spin systems) of infinitisimal size which conspire to evade experimental detection as magnetic dipoles.

The X-Y model of spin systems and B-S condition
The X-Y model is a sort of a generalization of the Ising model, i.e., instead of discrete ±1 spin values one place a spin rotor at each site which can point in any diretion in a two-dimensional plane. There is a theorem based on Landau symmetry breaking arguments that such 2-D systems are not expected to exhibit long-range order due to transverse fluctuations [13]. However, this 2-D system possess quasi-long range order in finite-size systems at very low temperatures. This is the so-called Berezinskii-Kosterlitz-Thouless (BKT) transition [21,22], marked by the occurrence of bound vortex-antivortex pairs the at low temperatures to unbound or isolated vortices and antivortices above some critical temperatures. In 1972, Kosterlitz and Thouless (KT) made a complete identification of a new type of phase transition in 2-D systems where topological defects in the form of vortices and antivortices play a crucial role. For their work they were awarded the Nobel Prize in 2016. Here we are mainly concern on how the B-S condition governs the nature of vortices of the X-Y model of spin systems. After the work of KT strong interest towards phase transition without symmetry breaking become mainstream. It was realized in passing that long before, the regular liquid-gas transition does not break symmetry, this was recognized without given much significance. Later on, Polyakov extended the work of KT to gauge theories, for example, 2+1 "compact" QED has a gapped spectrum in the IR due to topological excitations [23]. The SU(N) Thirring model has a fermions condensing with finite mass in the IR without breaking the chiral symmetry of the theory [24] . It was also further extended by in the context of 2-D melting of crystalline solids [25] leading to a new liquid crystalline hexatic phase. Another violation of Landau symmetry breaking arguments is exemplified by the Haldane phase transition [18].

8.1
The Hamiltonian for the X-Y model of spin system The X-Y model is a generalization of the Ising model, where Ising spins σ = ±1 are replaced by planar vector rotors of unit length, which can point in an arbitrary direction within the X-Y plane, The Hamiltonian is given by The cosine function can be expanded in powers of (ϑ i − ϑ j ), In the continuum limit, Eq. (14) leads to the following Hamiltonian studied by BKT, where −2JN ≡: E o is the energy of the system when all N spin rotors are aligned, and θ (r) labels the angle of the rotors at each point in the X-Y plane. The partition function in this continuum limt must account for all possible configurational function {θ (r)}. We are thus lead to a functional integral of the partition function, Z,

Remark 1
The X-Y model of spin system also serves as a beautiful model of Berry connection and Berry curvature in a more explicit and much simpler form. This naturally leads us to the B-S quantization condition for the vortex solutions given by KBT. It is therefore expected that this model will have an impact not only in statistical physics but also in quantum field theory and subject to generalizations. For example, if we write the unit spin vector by employing the Dirac ket and bra notations, we have which is the Berry connection. This site connection is indeed better appreciated by the way the sites are coupled in Eq. (13). This remark serves as advanced view, as it relates to the B-S quantization, of the discussions that follow.

Vortices as solutions
To simplify the calculation of the functional integral of Eq. (16), we use perturbative techniques which allows us to make use of the saddle point approximation. This entails expansion of the functional in terms of small fluctuations, δθ, around the minimum of H at θ o . Therefore, we need Then the field configurations are approximated, θ (r) θ o + δθ (r). We have for the partition finction, Here The minimum is determined from the solutions of Eq. (17), which correspond to solving the following equation, Although the analysis of the solutions to Eq. (18) proceeds classically, it has accurate analogy to quantum physics and contains the elements of Berry phase, Berry curvature and B-S quantization condition. Indeed, the X-Y model has some resemblance to the potential flow in two dimensional hydrodynamics 3 . There, the Laplace equation, ∇ 2 ϕ (x) = 0, is thoroughly analyzed for vortex solutions. Moreover, the X-Y model is reminiscent of the harmonic oscillator which also serves as a perfect classical forebear of the annihilation and creation operator, or ladder operators formalism, in quantum mechanics.
The two types of solutions to ∇ 2 θ (r) = 0, The solution given by The circulation velocities are given by These results are depicted in Fig. (2)-(3). The vortex solution θ + corresponds to a vortex charge +1, and θ − corresponds to antivortex charge −1. Both θ + and θ − are singular at (x, y) = (a, b).

B-S quantization of X-Y model vortex charges
For the X-Y model of spin system, the quantization of vortex charge, in contrast to the magnetic monopole charge, usually proceeds classically in a form of a simple boundary condition, the so-called circulation integral (borrowed from 2 -D hydrodynamics) belonging to the domain of integers, Z. Since the existence of a vortex is based on the existence of a singularity, the vortex charge can be singly charged, ±1 or multiply charged, ±m. The boundary condition for vortex solution is given by Eq. (19). The condition, C ∇θ · d l = 2πm basically follows from the single valuedness of n = cos θ sin θ . The number m ∈ Z is also called the winding number or vortex charge (by identifying with the potential problem in electrostatics, which is equal to the electric charge enclosed by the surface). Mathematically speaking, here our order parameter manifold is Only when there is a singularity would there be a point charge, or vortex core in the present instance given by 1 2π which is our B-S quantization analogy for the X-Y model.

Energetics of X-Y model
The energy cost of a single vortex has been shown to be, where L is the size of the system and a is the lattice constant. Whereas, the entropy of a votex is given by Thus the free energy cost due to presence of votex, which give the competition between order and disorder is The BKT transition occurs when ∆F =⇒ 0. This give the transition temperature, T KBT = πJ 2k B . For T < T KBT , the system is unstable against the formation of vortex-antivotex pair. The energy of the vortexantivortex pair is E pair = πJ ln r a where r is the separation of the pair. This is much less than the energy of a single vortex in the limit of large system size. For T > T KBT , the pairs become unbounded.

B-S quantization condition in Haldane phase
It is worth mentioning that quantization of the internal precession frequency of spin-1 chain soliton studied by Haldane [18], i. e., soliton angular momentum, S Z , follows that of Eq. (2). The gapped ground state of odd integer-spin chain is now called the Haldane phase, a new topological state of matter [19]. The Haldane phase, also called a symmetry protected topological state, is viewed as a short-range entangled which bears some similarity to the Kosterlitz-Thouless vortex solution of the the X-Y model of spin. We will not go into the details of Haldane phase as this will take us far from the scope of this paper.

B-S quantization condition in superfluids
For superfluds the B-S quantization condition is not implemented as a boundary condition but as a real B-S quantization condtion. Superfluid maybe viewed as a classical complex-valued matter field with emergent constant of motion, the topological order. Similar to the X-Y model, the complex matter field of Bose-Einstein condensate, ψ (r), is given by ψ (r) = |ψ (r)| e iΦ(r) Likewise the superfield velocity field is given by where γ = m is some constant. Here, the circulation is effectively quantized using the B-S quantization condition, i.e.,

B-S quantization of vortices in superconductors
A well-known property of superconductors is that they expel magnetic fields, the so-called Meissner effect. In some cases for sufficiently strong magnetic fields however, it will be energetically favorable for the superconductor to form a lattice of quantum vortices through the superconductor each which which carry quantized magnetic flux. A superconductor that is capable of supporting vortex lattices is called a type-II superconductor, vortex-quantization in superconductors is general.
The B-S quantization of vortices is a direct U (1) theory of connection, gauge or vector potential. We have, where φ o = hc e is the quantum flux. In the next section, we will discuss the B-S quantization condition in coherent state (CS) formulation of quantum mechanics. This will introduce us to the Berry phase and Berry connection in quantum field theory [29], specifically non-Abelian gauge theory of elementary particles, which is still an active field of research. For example, in effective quantum field theory, the Berry phase is equivalent to Wess-Zumino-Witten action for the sigma model [30].

B-S quantization condition in CS formulation of quantum mechanics
First we will treat the iconic harmonic oscillator in terms of ladder operators. This is the precursor to the CS formulation of quantum physics.

Harmonic oscillator
The Hamiltonian is, and the equations of motion areq We defined the ladder operators, i.e., annihilation and creation operators, α and α † as In terms of this operators, we have,

B-S quantization for harmonic oscillator
We will try to implement the B-S quantization condition by using the momentum and position operators in terms of the ladder operators and their eigenvalues.
The B-S quantization now simply reads

Coherent states and B-S quantization
Coherent state is defined to be the right eigenstate of the annihilation operatorâ, Sinceâ is non-Hermitian, α = |α| e iθ is complex, |α| and θ are the amplitude and phase of the eigenvalue. The conjugate state α| is the left eigenstate of the creation operatorâ † , this follows by taking the Hermitian conjugate of Eq. (22). The state |α is the so-called coherent state. The usefulness of coherent states is that they form a basis for the representation of other states. Coherent states can never be made orthogonal, although for well-separated eigenvalues α, they can be made approximately orthogonal. Moreover, the set of coherent states is overcomplete, in the sense that the set of coherent states form a basis but are not linearly independent, i.e., they are expressible in terms of each other. The nice thing is that the complex eigenvalue α is labeled by the classical (average) values of position and momentum in the following sense, Re α = 1 √ 2 α|Q |α , 2 Im α Re α = 1 2 α|P |α α|Q |α = α| P |α α| Q |α whereQ andP are the scaled canonical operators given by, Indeed, we have The counting of α states can be derived from the counting of states in (p c , q c ) phase-space, where q c = α| Q |α and p c = α| P |α , namely, 1 2π dq c dp c = 1 π d Re α d Im α ⇒ 1 π dα 2 (abbreviated).
Since α| P |α appears to be proportional to the Berry connection, we proceed to apply the B-S quantization as follows Equation (28) for the CS exactly reproduce Eq. (21) for the harmonic oscillator. Exactly the same results were obtained as a B-S approximation by Tochishita et al [31] written in their paper as

Other areas were B-S condition have been used
There are several instances that the B-S condition have been used, which are beyond the scope of this paper. For example, it has been used for one-dimensional Bogoliubov-de Gennes Hamiltonian. It may also have relevance to the so-called Fermionic oscillator in the 2-state system and it corresponding coherent state formulation. Its relevance and utility in the area of non-Abelian gauge theory is indeed not yet clear and lack focused investigation. In the following section, we will discuss in more details the B-S quantization of the integer quantum Hall effect in electrical conductivity, since our approach is entirely new.

Quantum Hall effect and Bohr-Sommerfeld quantization
Our new approach to integer quantum Hall effect make use of quantum superfield lattice Weyl Transform nonequilibrium Green's function (SFLWT-NEGF) formalism [10]. The calculation of integer quantum Hall effect is another good example where the IQHE conductance is directly proportional to the B-S condtion of Eq. (1), i.e., σ xy = e 2 h α ∆φ total 2π = α e 2 h n α (30) and enters in the following simple form, where e 2 h is the quantum conductance. Again ∆φ total is identified as the B-S contour integral undergoing quantization. This will be derived in this section since our new quantum transport approach to IQHE is not well known. It is also worth mentioning that for mesoscopic circuits with ballistic conducting channel and perfectly conducting leads, the conductance is simply equal to e 2 h , the quantum conductance per electron spin, i.e., n σ ≡ 1 [10].
In the nonequilibrium many-body Green's function technique, the principal quantities of interest are the "reduced" or single-particle correlation functions defined as where ψ H (1) and ψ † H (2) are the particle annihilation and creation operators in the Heisenberg representation, respectively. The indices 1 and 2 subsume all space-time indices and other quantum-label indices.

The Wigner distribution and density matrix operator
If we write the second quantized operator for the one-particle ( p, q; E, t)-phase space distribution function asf where λ label the band index and σ the spin index [here we drop the Heisenberg representation subscripts H for economy of indices], then upon taking the average we obtain particle distribution function ρ λλ σσ (p, q) , a generalized Wigner distribution function, where we employ the four-dimensional notation: p = ( p, E) and q = ( q, t). Equation (33) is indeed the lattice Weyl transform of the density matrix operatorρ as where the RHS is the lattice Weyl transform (LWT) of the density matrix operator, which is identical to the LWT of −iG < (1, 2).
Thus expectation value of one-particle operatorÂ can be calculated in phase-space similar to the classical averages using a distribution function, T r ρÂ := Â = p,q;λλ σσ A λλ σσ (p, q) ρ λ λσ σ (p, q) , clearly exhibiting the trace of binary operator product as a trace of the product of their respective LWT's. This general observation is crucial in most of the calculations that follows.
The Wigner distribution function f W ( p, q, t) is given by integrating out the energy variable, We further note that provides the major time dependence in the transport equation that follows.

Renormalized Bloch-electron Hamiltonian
In the following, let us consider the crystal-lattice effective Hamiltonian for energy band n in the presence of uniform electic field obtained through the use of LWT given by [10,11] and E o,n K is the energy band function for the band index n. We denote the Bloch state vector which is an eigenvector of the crystal momentum operatorP by | p and the Wannier state vector which is an eigenfunction of the lattice position operatorQ by | q . Of course the Bloch function is given by x| p and the Wannier function is given by x| q . We have, by suppressing the band indices, Observe that in the presence of electric field, the effective Hamiltonian depends on both lattice position operatorQ and time t through the vector potentialÃ (t) =Fct and the scalar potential V = −e F · q. Therefore in general, the LWT is expected to depend on the variables ( p, q; E, t).

Space and Time Translation Operators
Commutation properties of the time and space translation operators,T (t) andT ( q), respectively, suggest that in the presence of electric field, gauge invariant quantities that are displaced in space and time acquires Peierls phase factors [12]. For example, where q = 1 2 ( q 1 + q 2 ) , t = 1 2 (t 1 + t 2 ) .
Using the four dimensional notation: p = ( p, E) and q = ( q, t), the Weyl transform A (p, q) of any operator A is defined by where λ and λ stands for other discrete quantum numbers. Viewed as a transformation of a matrix, we see that the Weyl transform of the matrix q , λ Â q , λ is given by Eq. (41) and the lattice Weyl transform of p , λ Â p , λ is given by Eq. (42). Denoting the operation of taking the lattice Weyl transform by the symbol W then it is easy to see that the lattice Weyl transform of Similarly Using the form of matrix elements in Eq. (40), we have Hence the expected dynamical variables in the phase space including the time variable occurs in particular combinations of K and E. Therefore, besides the crystal momentum varying in time as the energy variable vary with q as In effect we have unified the use of scalar potential and vector potential for a system under uniform electric fields. The LWT of the effective or renormalized lattice Hamiltonian H ef f H ( p, q; E, t) can therefore be analyzed on K, E -space as The last line is by virtue of Eq. (46). And all gauge invariant quantities are functions of K, E such as the electric Bloch function [12] or Houston wavefunction [ [8]] and electric Wannier function, i.e., the electric-field dependent generalization of Wannier function. In particular, the Weyl transform of a commutator, where Λ is the Poisson bracket operator. We can therefore write the Poisson bracket operator Λ, as on K, E -phase space.

SFLWT-NEGF transport equation
The nonequibrium quantum superfield transport equation for interacting Bloch electrons under a unifrom electric field has been derived in Sec. VI of Buot and Jensen paper [12]. In the absence of superconducting behavior, the SFLWT-NEGF phase-space transport equation reads ∂ ∂t G < ( p, q; E, t) = 2 sinΛ H (p, q) G < (p, q) + Σ < (p, q) Re G r (p, q) If we expand Eq. (48) to first order in the gradient, i.e., sin Λ Λ,we obtain where G r (p, q) is the LWT of the retarded Green's function, A (p, q) is the spectral function, and Γ (p, q) is the corresponding scattering rate.

Ballistic transport and diffusion
We wiill simplify Eq. (49) by neglecting the self-energies, i.e., we limit to non-interacting particles. Then we have the following simplified quantum transport equation, which can be written in terms of the Poisson bracket of Eq. (47) as Therefore Assuming that the electric field is in the x-direction. Then The Hall current in the y-direction is thus determined by the following equation, If we are only interested in linear response we may consider all the quantities in the integrand to be of zero-order in the electric field, although this is not necessary if we allow very weak electric field leading to time dependence being dominated by the time dependence of the density matrix, as we shall see in what follows.

Topological invariant in (p, q; E, t)-space
From Eq. (51), we claim that the quantized Hall conductivity is given by and is quantized in units of e 2 h , i.e., σ yx = e 2 h Z, where Z is in the domain of integers or the first Chern numbers. In doing the integration with respect to time, t, we need to examine the implicit time-dependence of the matrix element of G < in the 'pull back' representation defined below.

SFLWT-NEGF approach to Hall conductivity
To prove that Eq. (52) gives σ yx = e 2 h n, where n ∈ Z, we need to transform the integral of the equation to the curvature of the Berry connection in a closed loop. This necessitates a 'pull back' (i.e., undoing or inverse) of the lattice transformation of Eq. (52).
The pull back procedure is founded on the observation that the proper phase-space integral of a product of the respective lattice Weyl transform of two operators is equivalent to taking the trace of the product of the same two operators in any choosen basis states of the system.

'Pull Back' of the lattice Weyl transformation
We give the proof that the Hall conductivity σ yx = e 2 h Z in Eq. (52) by recasting the Eq. (51) to the expression originally used by TKNN[ [3]], obtained through the Kubo formula in linear response theory, to derive the quantization of integer quantum Hall effect. This means we have to 'pull back' or undo the lattice transformation of SFLWT-NEGF transport equation. The details of the pull-back procedure or inverse LWT is given in the Appendix [32] The result of converting the quantum transport equation in the transformed space, Eq. (51), to the untransformed space by undoing or 'pulling back' the lattice Weyl transformation W, amounts to canceling W in both side of the equation given by, The time integral of the RHS amounts to taking zero-order time dependence [zero electric field] of the rest of the integrand, then we have for the remaining time-dependence, explicitly integrated as, Thus eliminating the time integral we finally obtain.
Taking the Fourier transform of both sides, we obtain  y (ω) Taking the limit ω =⇒ 0 and summing over the states β, we readily obtain the conductivity, σ yx .
where K =⇒P from Eq. (36) in both Eqs. (55) and (56). This is the same expression that can be obtained to derive the integer quantum Hall effect from Kubo formula [3].
We now prove that for each statevector, α, k , the expression, is the winding number around the contour of occupied energy-bands in the Brillouin zone. First we can rewrite the terms within the square bracket as The last term indicates the operation of the curl of the Berry connection which is related to the quantization of Hall conductivity. This quantization is due to the uniqueness of the parallel-transported wavefunction.
To understand this, we refer the readers to the Appendix which discusses how the phase of the wavefunction relates to the Berry connection and Berry curvature.

First Chern Number and Quantization of QHE
At low temperature, we can just write is the winding number or the Chern number. Therefore, over all occupied bands α, where n α ∈ Z is the topological first Chern (or winding) number. Thus the the Hall conductivity is quantized in units of e 2 h as derive from Eq. (52) of the new quantum transport approach used here.

Kubo current-current correlation formula
The way the B-S condition permeates the Kubo current-current formula is implicit and was utilized by TKNN to derive the topological anomalous (i.e., without magnetic field) IQHE.
To touch base with a time-dependent perturbation of the Kubo current-current correlation we recall that in this particular approach, a time varying electric field is indirectly used. To get to QHE the limiting case of ω =⇒ 0 is taken after Fourier transformation of a convolution integral. In adapting to our approach, this means that the time integral in the expression of the RHS of Eq. (53) when transformed to current-current correlation is a convolution integral before taking the Fourier transform.
We start with the RHS of Eq. (53), We make use of the general relations Similarly, we have Substituting in Eq. (60), we then have the convolution integral with respect to time, Consider the following Fourier transformation, We can transform the range of integration as follows, since f (α) = j (α) is the observable current density and hence self-adjoint. We can apply this result in what follows.
Defining the current density as j x = evx a 2 , we obtain, after Fourier transforming the convolution integral as where η is just a regularization exponent at ∞. Therefore the Kubo formula for the conductivity is given by This is the Kubo current-current correlation formula for the Hall conductivity.

Orbital magnetic moment, B-S condition, and edge states
The orbital magnetic moment is given by, Using Eq. (62), we obtain So Berry's curvature implies the presence of orbital magnetic moment. Using Eq. (62), we obtain as written in some literature ) .
In fact we can extract the Berry curvature by rewriting in the Heisenberg picture, From Eq. (37), we obtain Integrating the RHS with respect to time, the result is in dimensional units of an orbital magnetic moment multiplied by time denoted by M , The last line is just explicitly revealing the Berry curvature derived from the expression for the orbital magnetic moment. Equation (65) has the similar dimensional units of a Bohr magneton multiplied by time t. The resulting quantization of orbital motion leads to edge states for 2-dimensional systems.

Concluding remarks
In summary, we have shown that the old Bohr-Sommerfeld quantization conditon is really a theory of Berry connection or a U (1) gauge theory. We have demonstrated how it permeates and pervades the whole of modern quantum physics, implicitly identifying the B-S condition as the forebear of modern geometrical or topological quantum theory. In particular we have shown how it permeates the theory of the Nobel prize winning discoveries of quantum Hall effect, Kosterlitz-Thouless transition in two-dimensional systems, Haldane phase, and quantized vortices in superfluids. Here, we also show how the U (1) gauge theory naturally give us the quantized magnetic charge of Dirac monopole.
Our new real-time SFLWT-NEGF quantum transport approach have naturally lead us to B-S quantization of IQHE. We have identified topological invariant in ( p, q; E, t) −phase space quantum transport given Eq. (52), an integral expression which give results in Z manifold, the so-called first Chern numbers. Moreover, the conventional linearity in the electrical field strength may not be a necessary and sufficient condition to prove the integer QHE, but rather it is the first-order gradient expansion in the real-time SFLWT-NEGF quantum transport equation. In general, nonlinearity in the electric field is still present in the variable K, E in the integrand of Eq. (52) based on the assumption of weak electic field and hence weak time dependence. There seems to be experimental evidence on this nonlinearity [33]. It also appears that electron-electron interaction which does not break the symmetry of Eqs. (38)-(39) can be treated in similar manner.
The quantum transport approach, based on Buot's SFLWT-NEGF formalism [10], seems direct and natural for calculating the IQHE. It bypasses the use of Kubo formula for the current-current correlation which is based on time-dependent perturbation, thus the need to take the ω =⇒ 0 limit. To our knowledge, this is the first time that real-time SFLWT-NEGF quantum transport formalism is demonstrated to yield the topological invariants of condensed matter systems. The real-time SFLWT-NEGF multi-spinor quantum transport equations are also able to predict various entanglements leading to different topological phases of low-dimensional and nanostructured gapped condensed matter systems [34].
is the Bohr-Sommerfeld quantization condition, Eq. (1), reminiscent of the way Landau level degeneracy is calculated.
We can write explicitly showing the integal of Berry curvature in k-space.

Appendix B. 'Pull back' of the lattice Weyl transformation
We give the proof that the Hall conductivity σ yx = e 2 h Z in Eq. (52) by recasting the Eq. (51) to the expression originally used by TKNN[ [3]], obtained through the Kubo formula in linear response theory, to derive the quantization of integer quantum Hall effect. This means we have to 'pull back' or undo the lattice transformation of SFLWT-NEGF transport equation.
Consider the integrand in Eq. (52) given by the partial derivatives of lattice Weyl transformed quantities.
The trick is to 'pull back' (undo) the lattice Weyl transformation to touch base with Berry connection and Berry curvature. Take first the term of Eq. (66) From Eq. (44) this can be written as a lattice Weyl transform W in the form, where α, ∂ ∂ Kx K, E symbolically denotes derivative with respect to K x of the state vector α, K, E labeled by the three quantum labels. Likewise for β, ∂ ∂ Kx K, E . We also have whereρ is the density matrix operator. From Eq. (34), we take the time dependence of β, K, E (iρ) α, K, E to be given by i β, K, E ρ (0) α, K, E e iω αβ t .
We have 4 The density matrix operatorρ 0 is of the form, where the weight function is the Fermi-Dirac function, Similarly, Hence Shifting the first derivative to the right, we have For energy scale it is convenient to chose to use f (E α ) in the above equation, with the viewpoint that α-state is far remove from the β-state in gapped states, so that we can set f (E β ) 0. The case α = β is indeterminate so that by setting f (E β ) 0 renders the summation to be well-defined. Therefore Since it appears as a product of two Weyl transforms, it must be a trace formula in the untransformed or pulled back version, i.e., for the remaing indices α and β we must be a summation, Similarly, we have Therefore we obtain . Now the LHS of Eq. (51), namely Using the result of Eq. (67), we have where ω βα α, ∇ k p |β, p = α, p| v |β, p Likewise G < K, E = iW β, K, E ρ α, K, E Again since Eq. (68) is a product of lattice Weyl transform, it must be a trace in the untransformed version, i.e., a (2π ) For calculating the conductivity we are interested in the term multiplying the first-order in electric field. We can now convert the quantum transport equation in the transformed space, Eq. (51), to the untransformed space by undoing or 'pulling back' the lattice Weyl transformation W, which amounts to canceling W in both side of the equation given by, The time integral of the RHS amounts to taking zero-order time dependence [zero electric field] of the rest of the integrand, then we have for the remaining time-dependence, explicitly integrated as,  Taking the limit ω =⇒ 0 and summing over the states β, we readily obtain the conductivity, σ yx .
where K =⇒P from Eq. (36) in both Eqs. (55) and (56). This is the same expression that can obtained to derive the integer quantum Hall effect from Kubo formula [3].
We now prove that for each statevector, α, k , the expression, is the winding number around the occupied contour in the Brillouin zone. First we can rewrite the terms within the square bracket as The last term indicates the operation of the curl of the Berry connection which is related to the quantization of Hall conductivity. This quantization is due to the uniqueness of the parallel-transported wavefunction.