This work is concerned with the derivation of the Green’s function for Landau-quantized carriers in the Group-VI dichalcogenides. In the spatially homogeneous case, the Green’s function is separated into a Peierls phase factor and a translationally invariant part which is determined in a closed form integral representation involving only elementary functions. The latter is expanded in an eigenfunction series of Laguerre polynomials. These results for the retarded Green’s function are presented in both position and momentum representations, and yet another closed form representation is derived in circular coordinates in terms of the Bessel wave function of the second kind (not to be confused with the Bessel function). The case of a quantum wire is also addressed, representing the quantum wire in terms of a model one-dimensional $\delta (x)$-potential profile. This retarded Green’s function for propagation directly along the wire is determined exactly in terms of the corresponding Green’s function for the system without the $\delta (x)$-potential, and the Landau quantized eigenenergy dispersion relation is examined. The thermodynamic Green’s function for the dichalcogenide carriers in a normal magnetic field is formulated here in terms of its spectral weight, and its solution is presented in a momentum$/$integral representation involving only elementary functions, which is subsequently expanded in Laguerre eigenfunctions and presented in both momentum and position representations.

## I. INTRODUCTION

Starting with the discovery of the exceptional electronic transport and detection properties of graphene, there has grown up an extremely broad and powerful research effort directed at understanding and developing the basic science, engineering and production as such Dirac-like materials, having a relativistic-type energy spectrum. These materials include silicene, topological insulators and now group-VI dichalcogenides. As an agent for probing the properties and also inducing new physical features, the magnetic field has always had an important role. Such studies of the effects of the magnetic field in nonrelativistic carrier propagation and collective modes date back to the 1960’s and 1970’s for three and two-dimensional quantum plasmas.^{1,2} The relativistic Green’s function in a magnetic field was most elegantly first determined by Schwinger,^{3} and interesting variants for particular Dirac-like materials have been advanced by Rusin and Zawadski,^{4} as well as the author.^{5} This paper is focused on yet another important representation for the Landau-quantized Green’s function for the group-VI dichalcogenides, particularly in direct time representation, whose behavior is more transparent in the low field limit in comparison with that of the eigenfunction expansion. Moreover, it facilitates the analysis of the role of a quantum wire, modeled here by a one-dimensional $\delta (x)$-potential profile, enabling the explicit determination of the wire Green’s function and its Landau quantized Dirac-like spectrum.

As in the case of graphene, the group-VI dichalcogenides have two Dirac points, *K* and $K\u2032$, at which relativistic-type spectra prevail for low energies. However, they also exhibit an energy gap and strong spin-orbit coupling, which must be taken into account. An appropriate **k** $\u22c5$ **p** model Hamiltonian is block diagonal, involving four $2\xd72$ blocks (each block for a specific choice of spin and valley), and the individual $2\xd72$ blocks are quite similar to that of graphene. Specifically, for the case at hand, the hamiltonian is given by

where inclusion of the valley index $\nu $ represents two of the four 2x2 blocks, and

is the canonical momentum $(\u210f=c=1)$. **A** is the vector potential of the normal magnetic field **B**, $\sigma =(\sigma x,\sigma y,\sigma z)$ are the Pauli spin matrices, $\gamma v=\gamma sign(\nu )$ with $\nu =\xb11$ as valley index and $\gamma $ is determined by the tight binding hopping parameter and lattice spacing. Furthermore, a spin index $sz=\xb11$ enters into energy shifts as $Esz=sz\nu \lambda 2$ and $g=\Delta 2\u2212Esz$, where $\lambda $ is the spin splitting and $\Delta $ is the energy gap without spin splitting.

Addressing the issue of gauge, it is useful to note that for a uniform constant magnetic field, the Green’s function equation [*I*_{2D} is the 2*D* unit matrix]

involves Hamiltonian dependence on the vector potential **A** (and magnetic field **B** directly through spin), $H(\u2207,A,B)$, which may be gauge transformed so that it is supplanted by $A(r)\u219212B\xd7(r\u2212r\u2032)$, with the corresponding gauge transformation of the Green’s function producing a Peierls phase factor^{1,3}$C(r,r\u2032)\u2009(R=r\u2212r\u2032;T=t\u2212t\u2032)$

and for the case of retardation, $G(r,r\u2032;t,t\u2032)\u22610$ for $t<t\u2032$.

In this, $C(r,r\u2032)$ embodies all dependence on $(r+r\u2032)$ if there is no other spatial inhomogeneity, $\varphi (r)$ is an arbitrary gauge function and $G\u2032(R,T)$ is translationally invariant in both position space and time, obeying the modified Green’s function equation

In addition to $G\u2032(R,T)$ being translationally invariant, it is independent of gauge. It will be analyzed here in rectangular position-time coordinates and in circular coordinates, also in momentum representation.

In Section II, the retarded Green’s function for group-VI dichalcogenides in a magnetic field is explicitly determined in a closed form integral representation in terms of elementary functions in both position and momentum representations; also its eigenfunction representation is derived. In Section III, the Green’s function for this system with a superposed model quantum wire $\delta (x)$-potential is also determined explicitly for propagation directly along the wire, and its energy spectrum is examined. Furthermore, the thermodynamic Green’s function and spectral weight for the Landau-quantized dichalcogenides is discussed in Section IV. In all cases we exhibit the Green’s function in both position and momentum representations. A summary is presented in Section V. Details of solution, integration and the dispersion relation are included in three appendices.

## II. LANDAU QUANTIZED GREEN’S FUNCTION

Considering the Hamiltonian of Eq. (1) jointly with Eqns. (4) and (5), the $2\xd72$ matrix Green’s function equation for $G\u2032(r\u2212r\u2032;t\u2212t\u2032)$ near the $K,K\u2032$ points of the group-VI dichalcogenides may be written as

or, in frequency representation (note that we discuss the *retarded* Green’s function here, $\omega \u2192\omega \u2009+\u2009i0+$):

where

The 4 elements of the Green’s function matrix equation (Eq. 7) are given by

Employing Eqns.(10) and (12) to eliminate $G21\u2032$ and $G12\u2032$ from Eqns.(9) and (11) for $G11\u2032$ and $G22\u2032$, respectively, we have

where $G1122\u2032\u2009=\u2009G11\u2032$ or $G22\u2032$ corresponding to the upper or lower of alternative $\xb1$,$\u2213$ signs elsewhere in the equation.

On the basis of Eq. (8) and the identity

we obtain

This result may be further simplified by noting that the operator $LZ$,

vanishes in application to the Green’s function,

because of conservation of the *z*-component of angular momentum. (The creation of a particle at ($x\u2032,t\u2032$) provides angular momentum which is subsequently eliminated with its annihilation at (** x**,

*t*); this is readily verified by expansion of $G(x,t;x\u2032,t\u2032)$ in angular momentum eigenfunctions.) Moreover taking the arbitrary gauge function $\varphi \u22610$, $LZ$ commutes with $C(x,x\u2032)$ so Eq. (17) reads as

Correspondingly, we have

or

with the definitions

The solution may be written in a “time”-$\tau $ representation as ( Appendix A)^{1}

and noting that the $\tau $-integrand is the generator of Laguerre polynomials *L*_{n}, we obtain^{6} (notation: $\gamma \nu \u22611\nu \gamma $ with $1\nu =\xb11$ for $KK\u2032$Dirac points)

In the absence of the energy shifts presently under consideration, Dirac-like Landau-quantized eigenenergies of the form for graphene are given by $\mathit{\epsilon}n\xb12=(2n+1\u22131\nu )\gamma 2eB$, so we write the energy denominators on the right of Eq.(23) as

Correspondingly, the eigenenergy spectrum for the case at hand is given by $Dn\xb1=0$, whence ($\u210f=1$)

With the results of Eq.(23) for $G11\u2032$ and $G22\u2032$, the determination of $G21\u2032$ and $G12\u2032$ can be readily carried out using Eqns. (10) and (12), which may be summarized as

Using the identity^{7}

($Lnm(z)$ are generalized Laguerre polynomials) we have

Thus, $G21\u2032(R;\omega )$ is given by

where

with a corresponding adjoint relation for $G12\u2032(R,\omega )$ (involving the interchange of positional matrix elements $r\u2194r\u2032$ so that $R\u2192\u2212R$, as well as complex conjugation).

For many purposes^{8} it is preferable to use momentum **p**-representation rather than position representation. Fourier transforming Eq.(22), we obtain $R\u2192p$, Appendix A)

It is also of interest to examine Eq.(20) in circular coordinates $(R,\mathit{\theta})$:

(the factor of 2 in parentheses on the right compensates for the semi-infinite range of $R=[0\u2192\u221e]$). Actually, Eq.(23) exhibits $G1122\u2032(R,\omega )=G1122\u2032(R,\omega )$ as a function of radius *R* alone, to the exclusion of angle $\mathit{\theta}$. Consequently, $\u22022G1122\u2032(R,\omega )/\u2202\mathit{\theta}2\u22610$and the angular average $\u222b02\pi d\mathit{\theta}2\pi \u2026$ of Eq.(31) yields

This result has the form of a Bessel Wave Equation of order 0 for all $R>0$ (with $\delta (R)=0$), and its homogeneous solutions are linear combinations of Bessel Wave Functions^{9} $Z1(R)$ of the first kind and $Z2(R)$ of the second kind. For an infinitesimal integration range of $R\u2192[0,\mathit{\epsilon}]$, it behaves as

so cancellation of the common $1R$-factor and integration $\u222b0\mathit{\epsilon}dR\u2026$ yields

which corresponds to

This behavior for small $R\u223c\mathit{\epsilon}$ is indicative of the $Z2$ second Frobenius solution of the Bessel Wave Equation of order 0, whose leading term is $ln\mathit{\epsilon}$. Accordingly, we indentify this solution

as an alternative to Eq.(30).

## III. QUANTUM WIRE

This section is focused on Green’s function dynamics of a quantum wire modelled by a 1D Dirac delta function $\delta (x)$-potential well profile superposed on carriers in a 2D group-VI dichalcogenide material subject to a normal quantizing magnetic field. This additional potential ($\beta <0$ is the product of the potential well depth and its width)

modifies the hamiltonian in Eq.(1) above, and the associated matrix Green’s function *G*^{W}-equation can be written in integral form as ($\omega $-representation)

or ($G(r1,r2;\omega )$ is the Green’s function in the absence of the wire; suppress $\omega $)

As in the case of *G*, gauge considerations mandate the presence of a Peierls phase factor in *G*^{W}:

However, $G\u2032W(r1,r2;\omega )$ lacks translational invariance in the *x*-direction because of the $\delta (x)$-wire potential. This complicates the full determination of $G\u2032W$ for off-axis dynamics, but if propagation is restricted to the wire itself, $x\u22610$ (*x*_{1} = *x*_{2} = 0), we have

Furthermore, $G(0,y1;0,y3)\u2009=\u2009G\u2032(0,y1;0,y3)$ since $C(r1,r3)\u22611$ for *x*_{1} = *x*_{3} = 0; and both $G\u2032$ and $G\u2032W$ are both manifestly translationally invariant in the *y*-direction. Accordingly, this integral equation is readily solved using a single Fourier transformation in the *y*-direction,

with the result (on *y*-axis, $x\u22610$)

for propagation along the wire. A similar result obtains for propagation strictly parallel to the wire since the Peierls phase factor is unity when **r**_{1} and **r**_{2} are parallel, $C(r1,r2)=exp(ie2r1\u22c5B\xd7r2)\u22611$. Alternatively expressed in *p*_{y}-representation Eq.(41) may be expressed in a more symmetric form using *G*^{W}(*p*_{y}) from Eq.(43), as

The infinite-space Green’s function involved in the description of propagation along the wire, $G(py)\u2261G(py,x\u22610)$, is given in terms of its elements as follows:

For the diagonal elements,

where $Dn\xb1$ was defined in Eq.(24). Expanding $Ln(m\omega c[X2+Y2]/2)$ in products of Hermite polynomials with *X* = 0 yields^{10}

and the *Y*-integral can now be carried out using related Hermite polynomials $Hen(x)\u22612n/2Hn(x/2)$, with the result ( Appendix B)^{11}

where we have defined

In regard to the off-diagonal elements, their relation to the diagonal elements in *p*_{y}-momentum representation (with $X\u22610$) may be written as

or

The wire energy spectrum in the case of a thin wire (small $\beta $) requires $Dn\xb1$ to also be small–of order $\beta $– in order to satisfy the determinental relation

This is to say that a root determined by $Dn+\u2009\u2245\u2009O(\beta )$ involves $\beta G11\u2032(py)\u2009\u2245\u2009O(\beta 0)=O(1)$; but *D*_{n−} and *D*_{n+} differ by order $O(\beta 0)=O(1)$ so *D*_{n−} is correspondingly of order $Dn\u2212=O(\beta 0)=O(1)$ and $\beta G22\u2032(py)\u2009\u2245\u2009O(\beta )$ is negligible. Similar considerations apply when ”+” and ”−” are interchanged along with the interchange of $G11\u2032$ and $G22\u2032$, and they also confirm the negligibility of $\beta 2G11\u2032G22\u2032\u2212G21\u2032G12\u2032$ for small $\beta $ ( Appendix C). In this case, an energy root $\omega \u2213$ is sufficiently close to the nearest pole position so that its location is determined by that pole alone, with the result ($\omega \u2192\omega n$ for the $nth$ root; note that $\xb1\u2032$ indicates an additional proliferation of modes, some of which may be extraneous):

or

## IV. SPECTRAL WEIGHT AND THERMODYNAMIC GREEN’S FUNCTION

The thermodynamic magnetic field matrix Green’s function for the group-VI dichalcogenide materials also satisfies Eqns.( 1-19) above, and it will be distinguished by an overhead bar, bold letter $G\xaf$,

with $C(r,r\u2032)$ as the usual Peierls phase factor of Eq.(4). This thermal Green’s function $G\xaf(r,r\u2032;T)$ in direct time representation has *non*vanishing parts $G\xaf>(r,r\u2032;T)$ and $G\xaf<(r,r\u2032;T)$ for *both*$T>0$ and $T<0$, respectively $\eta +(T)=0,1\u2009for\u2009T<0,T>0$, respectively, is the Heaviside unit step function):

A spectral weight matrix **A**(*T*) operating in the same spectral pseudospin subspace as $G\xaf$ may be defined in direct time representation as

and, equivalently,

with

Since both $G\xaf\u2032>(R;T)$ and $G\xaf\u2032<(R;T)$ satisfy the homogenous counterpart of the Green’s function equation (absent the driving term $\delta (2)(R)\delta (T)$ with *T*$\u2260$ 0), so does the spectral weight function $A\u2032(R,T)$. The latter is subject to the sum rule (due to the canonical commutation/anticommutation relations)^{1,12,13}

where the spatial delta function mandates that $r=r\u2032inC(r,r\u2032)\u2192C(r,r)\u22611$.

In frequency representation, the spectral weight function (matrix) provides the ”greater” $(G\xaf>)$ and ”lesser” $(G\xaf<)$ parts of the thermodynamic Green’s function, corresponding to $t>t\u2032$ and $t<t\u2032$, respectively, as^{12,13}

where $f(\omega )$ is the Fermi (or Bose) statistical distribution function ($\xb1$ here refers to bosons (upper sign) or Fermions (lower sign)). As it is often preferable to work in **p**-momentum representation, we consider the $A11\u2032(p,T)$ and $A22\u2032(p,T)$ elements of $A\u2032(p,T)$, which must satisfy the homogenous counterpart of the Green’s function equation

subject to the sum rule Eq.(58) in momentum representation

and from Eq.(59) with $R\u2192psett\xaf=\Omega c\xb1T/2$ ,

It is immediately clear that the periodic $cos(t\xaf)$ and $tan(t\xaf)$ functions in Eqns.(62, 63) may be written in a Fourier series, producing a row of frequency $\delta $-functions in $\omega $-representation, which will place the eigenmode energies in the arguments of the statistical $f(\omega )$ -functions in Eq.(63). This may be seen from yet another point of view, since the spectral weight matrix function $A(r,r\u2032;\omega )$ can also be determined from the *retarded* Green’s function $G(r,r\u2032;\omega )$ discussed above as^{13}

Employing Eq.(23) for $G\u2032(R,\omega +i0+)$ we obtain the diagonal elements of $A1122(r,r\u2032;\omega )$ as

In dealing with the position/momentum representations of the thermal Green’s function and spectral weight as $2\xd72$ pseudospin matrices, it is should be borne in mind that the important spectral weight function trace $A\u2032$ in diagonal energy representation is given by the collection of *all* energy state contributions, and since the trace is invariant with respect to change of basis/representation, it is equally well given in the pseudospin position/momentum matrix representations by the trace of Eqns.(65,66) whence

excluding the off-diagonal elements. With this, the trace of Eq.(59) provides the associated thermal Green’s function trace, subject to the addition of the other spin/valley contributions.

## V. CONCLUSIONS: SUMMARY

This work has addressed the matrix Green’s function *G* for group-VI dichalcogenide materials in the presence of a normal, quantizing magnetic field. After seperating the Peierls phase factor, the translationally invariant part of the retarded Green’s function $G\u2032$ is derived in position/frequency representation in a closed form integral representation involving only elementary functions, Eq.(22) (diagonal elements, from which the off-diagonal elements are easily determined by Eqns.(10,12)): This integral representation, Eq.(22), is also expanded in an eigenfunction series of Laguerre polynomials, explicitly exhibiting both the diagonal (Eq.(23)) and off-diagonal (Eqns.(28,29)) elements as functions of position and frequency and identifying the energy eigenvalues (Eq.(25)). It is also of interest to present the retarded dichalcogenide Green’s function (diagonal elements) in direct time *T*-representation (as opposed to the $\tau $ -”time” transformation variable of convenience employed in Eq.(22) as conjugate to $\Omega \u2213$): This is given by Fourier $\omega $-integration of Eq.(23) as

As momentum representation is of great importance in more complicated calculations, the closed form integral representation of $G\u2032$ is also determined as a function of momentum and frequency (Eq.(30)). Yet another representation of $G\u2032$ is derived in circular coordinates in terms of the second Frobenius solution of the Bessel wave equation of order 0 in Eq.(36).

This work has also addressed the presence of a quantum wire impressed on the dichalcogenides by a model one-dimensional $\delta (x)\u2212$potential profile. The associated retarded Green’s function integral equation for this system is solved exactly for propagation directly along the wire (Eqns.( 44-50)) in terms of the corresponding Green’s function for the system with*out* the wire $\delta (x)\u2212$potential, and the eigenenergy dispersion relation is determined (Eq.(51)) and solved approximately (Eq.(53)).

We have also formulated the thermodynamic Green’s function (matrix) for the group-VI dichalcogenides in terms of its spectral weight, exhibiting its solution in a momentum/frequency integral representation in terms of elementary functions alone (Eq.(63)); moreover, using a Laguerre polynomial expansion, it is also presented in both position/frequency and momentum/frequency representations (Eqns.(65,66)). (Important mathematical issues relating to the solution and integrations are presented in Appendices A and B, respectively.)

In all representations considered here, for both the homogeneous infinite system and the quantum wire model as well, the principal affect of the magnetic field is the splintering of the energy spectrum into a proliferation of discrete Landau-quantized energy states, as was the case for graphene and a quantum dot model study done previously.^{5} Of course, the determination of the Green’s functions in these cases also provides detailed information about the pertinent carrier propagation characteristics, including the Peierls phase factor and orbit curvature, as well as the energy eigenstate spectra.

## ACKNOWLEDGMENTS

Prof. M.L. Glasser has been helpful in resolving some mathematical issues that arose in the course of this work.

### APPENDIX A: SOLUTION

and returning to a ”time”-$\tau $ representation we have ($\tau $ is a ”time” variable conjugate to $\Omega \u2213$ under Fourier transformation)

Here, $\Omega \u2213$ is viewed as the conjugate frequency variable and

$G1122\u2032(px,py,\tau )$ obeys the homogeneous counterpart of Eq.(B) for $\tau >0$$(\delta (\tau )=0)$ and the homogeneous solution has the form^{1} (*k*_{1},*k*_{2} are constants)

Integrating Eq.(B) from $\tau =0\u2212$ to $\tau =0+$, the initial condition for retardation requires that $G1122\u2032(px,py;\tau =0+)=\u2212i$, whence we obtain

and

Fourier transformation to position representation yields $(px,py\u2192X,Y)$

Here, we note that inverting the Fourier transform from $\tau \u2212$representation to $\omega $-representation, $G1122\u2032(R,\Omega \u2213)$, returns us to the original physical quantity represented by the function $G1122\u2032(R,\Omega \u2213)=G1122\u2032(R,\omega )$(although mathematicians would write it differently).

### APPENDIX B: INTEGRALS

#### A. The integral $I1=\u222b\u2212\u221e\u221edYe\u2212ipyYexp\u2212eB4Y2H2keB2Y:$

Introducing the related Hermite polynomials $Hen(x)=2n/zHnx2$, and changing the scale of the integration variable $x\xaf\xaf=eB2Y$, we have

#### B. The integral $I2=\u222b\u2212\u221e\u221edzRe\u2212ip\u22c5Re\u2212eB4R2LneB2X2+Y2:$

The Laguerre function may be expanded in Hermite polynomials separating the variables of integration as^{10}

whereupon the double integral separates completely:

These one-dimensional integrals can be evaluated using Eqns.(B1, 2) above and the resulting sum of products of Hermite polynomials can be re-assembled into the Laguerre polynomial using Eq.(B3), with the result

### APPENDIX C: DISPERSION RELATION ($\beta $ SMALL)

The determinental dispersion relation takes the form

which may be written out more fully for small $\beta $ as

For $\beta $ small, then a denominator *D*_{n+} or *D*_{n−} must be order $O(\beta )$ to obtain a root. However, *D*_{n+} and *D*_{n−} differ by order $O(\beta 0)\u2009\u2245\u20091$, so if one is of order $O(\beta )$ then the other is of order $O(\beta 0)\u2009\u2245\u20091$, and the term associated with the latter is smaller than that of the former because its denominator is larger by a factor of $\beta $. Extending these considerations to the last term on the right involving the denominator product $Dn+Dn\u2212\u223cO(\beta )$ shows that the last term is of order $O(\beta 2/\beta )\u2009=\u2009O(\beta )$, and it can be neglected for small $\beta $.