Green's functions and DOS for some 2D lattices

In this note we present the Green's functions and density of states for the most frequently encountered 2D lattices: square, triangular, honeycomb, kagome, and Lieb lattice. Though the results are well know, we hope that their derivation performed in a uniform way is of some pedagogical value.


I. INTRODUCTION
Fermionic lattice models are widely used not only as a purely theoretical tool but also as a basis for investigation and modelling of physical properties of real materials [1]. Despite their relative formal simplicity -the Hamiltonians of many of them can be written down as bilinears of fermionic operators -analytical calculation of the lattice Green's functions can present substantial difficulties.
In general, the lattice Green's functions of systems are ubiquitous [2][3][4] in solid state physics, appearing in problems of lattice vibrations, spin wave theory of magnetic systems, localized oscillation modes at lattice defects, combinatorial problems in lattices [5], and flux calculations in lattice percolation [6]. Lattice Green's functions are also central to the theory of random walks on a lattice [7,8], and to the calculation of the effective resistance of resistor networks [9].
The lattice Green's functions are of central importance for understanding the electronic behavior of perfect crystalline solids. They also provide the basis for understanding the electronic properties of real, imperfect crystalline solids, since the imperfections can be treated as a perturbation. Lattice Green's functiona are also important for calculating RKKY interaction [10][11][12][13].
We will consider below several popular 2D lattices and calculate Green's functions and density of states (DOS). The results are well known, still we think that derivation of these results performed in a uniform way and being presented in one place is of some pedagogical value.
In all cases we'll consider the models with the nearest neighbour hopping only, the amplitude of the hopping we'll take to be 1, so the Hamiltonian will be where c † and c are electron creation and annihilation operators, and the summation in Eq. (1) is with respect to nearest neighbor pairs. The period of the lattice we'll also take to be equal to 1.

II. SQUARE LATTICE
In the wavevector representation the Hamiltonian for the square lattice is The site diagonal matrix element of the Green's function is Employing the identity to perform integration with respect to k y , we obtain for E < −4 or E > 4 where a > b > c > d, K is the complete elliptic integral of the first kind, and In our case a = 1, b = −1; hence we obtain Thus we get for E > 4 We can analytically continue the Green function from the part of real axis E < −4. For −4 < E < 4 we have k > 1. Hence we should put E = ǫ + i0 + , that is k acquires infinitesimal imaginary part i0 + , and we may use the identities [4] Im Re to get [4] ρ The DOS is presented on Fig. 1

III. TRIANGULAR LATTICE
Let a 1 , a 2 and a 3 are the three periods of the triangular lattice adding up to zero. The Hamiltonian in the wave vector representation is where k i ≡ a i · k. The site diagonal matrix element of the Green's function is (k 1 and k 2 are the components of k in the oblique coordinate system) (we used the transformation suggested in Ref. 8). Employing the identity (4) to perform integration with respect to k 2 we obtain for E > 6 or E < −2 where [15] For E > 6 we may use the identity [14] c where a > b > c > d, and From Eq. (16) follows where Thus we get for E > 6 [16,17] We can analytically continue the Green function from the part of real axis E > 6. For −2 < E < 6 we have k > 1. Hence we should put E = ǫ + i0 + , that is k acquires infinitesimal imaginary part i0 + , we may use the identities (10), (11) and take into account that For −3 < E < 2 the value of k become imaginary. We may use the identity [18] where In our case and we reproduce DOS for the triangular lattice [16] ρ where The DOS is presented on Fig. 2.

IV. HONEYCOMB LATTICE
The honeycomb lattice can be considered as a triangular lattice with a basis of two lattice points.The tight binding Hamiltonian for the electrons iŝ where δ 1 , δ 2 , δ 3 are the vectors connecting an atom with its nearest neighbors. Going to wave vector representation we obtain Taking into account that δ 1 − δ 2 = a 3 , δ 2 − δ 3 = a 1 , δ 3 − δ 1 = a 2 , we obtain the spectrum as where ν = ±1, and The dispersion law can be written in the form The Green's function is .
Thus we obtain For the density of states we obtain Actually, Eqs. (36) and (37) become completely obvious after we square the Hamiltonian (31) to get The DOS is presented on Fig. 3

V. KAGOME LATTICE
The kagome lattice can be considered as a triangular lattice (which we consider to be identical to the lattice introduced above for the honeycomb lattice) with a basis of three lattice points. The Hamiltonian iŝ where δ i = a i /2 are the vectors connecting the nearest neighbors.
Going to wave vector representation we obtain The spectrum is found from the equation Calculating the determinant and taking into account that k 1 + k 2 + k 3 = 0 we may present the dispersion equation as Thus we have a flat band ǫ F (k) = −2, and two dispersive bands The Green's function is Comparing Eqs. (45) and (14) we obtain Hence We see, that apart from the δ-function peak, the DOS for the Kagome lattice is the DOS for the honeycomb lattice shifted by 1. The DOS is presented on Fig. 4  The spectrum of the Hamiltonian (48) is The Green's function is Comparing Eqs. (50) and (3) we obtain Hence The DOS is presented on Fig. 5

VII. CONCLUSIONS
To summarize, we have expressed lattice Green's functions and density of states for triangular and square lattice through the complete elliptic integral of the first kind in a uniform way. The Green's functions were calculated analytically as functions of energy on the part of real axis, and then continued analytically on the whole complex plane.
We connected lattice Green's functions and density of states for honeycomb and kagome lattices with those of the triangular lattice, and Green's functions and density of states for Lieb lattice with those of square lattice. Exact expressions for lattice density of states ought to be useful in dynamical mean field theory calculations [19].
We have also shown that the well known results for the wave functions of electrons in the vicinity of the Dirac points in the honeycomb lattice are not connected with the nearest-neighbor hopping approximation, but follow from the symmetry of the model.
The results for the wave functions of electrons in the vicinity of the Dirac points in the honeycomb lattice (Eq. (A9) and (A10) below) are well known [20]. In this appendix we want to emphasise the fact that these results are not connected, as it is sometimes erroneously stated, with the nearest-neighbor hopping approximation, but follow from the symmetry of the model. Looking at Eq. (33) we understand that to find minimal value of X k we should find minimal value of the sum of cosines of three numbers which add up to zero. This value is equal to −3/2 and is achieved when two of the numbers are 2π/3 and the third number is −4π/3 (the point K), or when two of the numbers are −2π/3 and the third number is 4π/3 (the point K ′ ). The bands touch each other at these points.
The wave functions can be presented as where Making substitution expanding with respect to q and keeping only the linear terms we obtain Thus we have conic point in the spectrum.
In the same approximation Hence θ q = θ q , where θ q is the polar angle of q, the X axis being chosen in the direction of n. If we chose then X axis is in the direction of δ 2 . Hence Eq. (A1) can be written as For the K ′ point, the wavefunctions are obtained from those in Eq. (A9) by permutation of the sublattices The natural question appears: are Eqs. (A9) and (A10) connected to the simplest possible tight-binding model we have used? The answer is: No, these equations are general, and follow from the symmetry of the problem. In fact, the general tight-binding Hamiltonian for the honeycomb lattice iŝ where the summation is with respect to all lattice vectors a, and δ is some arbitrary, but fixed δ i . The selection rule for matrix elements [21] gives a t(a + δ)e iK·(a+δ) = 0.
In fact, we are dealing with the product of two functions. The function t(a + δ) realizes the unit representation of the point symmetry group C 3 (the full symmetry group of the inter-sublattice hopping is C 3v , but the restricted symmetry is enough to prove the cancelation). As far as the function e iK·(a+δ) is concerned, rotation of the lattice by the angle 2π/3, say anticlockwise, is equivalent to rotation of the vector K in the opposite direction, that is to substitution of the three equivalent corners of the Brilluoin zone. Thus the exponent e iK·(a+δ) realizes x − iy representation of the group C 3 . Because each of multipliers in Eq. (A12) realizes different irreducible representation of the symmetry group, the matrix element is equal to zero. Simply speaking, at a point K the sublattices become decoupled, and this explains the degeneracy of the electron states in this point (these points) or, in other words, merging of the two branches of the single Brilouin zone.
Hence, for a general tight-binding model, the nondiagonal matrix element of the Hamiltonian (A11) is a linear function of q: (c + id) · q, invariant with respect to rotation of q by 2π/3 up to a q independent constant. It immediately leads to the demands |c| = |d|, and c ⊥ d and we recover Eq. (A9). Equation (A10) follows from Eq. (A9) because of invariance of the system with respect to mirror reflections.