In source code 1 see Appendix , we have illustrated how to obtain the graph of Fig. For an introductory course in Julia , please see Ref. Let us now apply the equation-of-motion technique to a linear chain composed of three sites, shown in Fig. Although it may appear as just another application of Eq. For the widely-used surface Green's function , this 3-site model is revisited briefly, however special attention is required by the surface-bulk method that will be presented. The visualization of the sites may help writing of the equations of motion, making it easier and mechanic.
The equations of motion of this system will play an important role for the recursive methods presented later on. From the local potential term of the Hamiltonian above, we see that the undressed Green's functions 66 can be written as. We now will write the EOM for the dressed Green's function , and for the non-diagonal propagators that connect the sites i and j. Green's function of site 1 : G 11 — Let us now calculate the Green's function of the first site of the three-site system, according to Eq. Schematically we see in Fig. One way of visualizing how it works is first to identify the first neighbor of the site in question see Fig.
Using the result of Eq. Green's function of site 2 : G 22 — Applying the practical scheme discussed above we can write an expression for the central Green's function as. These expressions are inserted into Eq. Green's function of site 3 : G 33 — The equation of motion for the local dressed Green's function of site 3 gives us. The resulting expression is. So far these examples not only provided us the opportunity to exercise the method but also introduced the boxed expressions 87 , 93 e 97 , fundamental to the technique developed in Sec.
In early 80's, the investigation of surface and bulk properties of metals, transition metals and semiconductors motivated the development of effective Hamiltonians and iterative techniques to obtain the density of states [ 42 ]. The recursive Green's functions RGF used computationally efficient decimation techniques from the numerical renormalization group, simulating materials via effective layers [ 43 ]. The success of recursive Green's functions was boosted by simulation of transport in materials, in particular in two-terminal ballistic transport. The retarded and advanced Green's functions of the central device in a junction contain information to the calculation of transport properties such as the stationary current and conductivity, or transmission matrix.
In essence, the idea of dividing the material in layers, modelling it in a chain, is the spirit of the recursive Green's function method. We will illustrate this procedure using a linear chain of single-site orbitals and two forms of decimation: the most widely-used, the surface technique, and an alternative version that stores information from the central sites. Let us consider a three-site chain, as shown in Fig. We will basically follow the references [ 43 , 44 ] except for the fact that in our notation, the first site is labelled as 1 instead of 0, therefore every index will be shifted by one with respect to the ones in [ 43 , 44 ].
Again, for the first site we have the equations of motion. The idea is to keep the three-site chain by renormalizing the hoppings and local energy of the first site. By replacing in 99 , we eliminate the non-diagonal propagator G 21 :. As a general rule, the non-diagonal propagator G n 1 relates first neighbors:.
Starting from G 11 , Eq. The first iteration is Eq. Next, the non-diagonal G 31 relates sites 1 and 5, and so on, as follows:. These equations except for the first one are analogous to the first-neighbors recursion, Eq. Starting from Eq. At each repetition we will obtain a larger effective system with not twice, but 2 x the lattice constant.
This process is known as decimation , where one encapsulates the numerous sites into a three-point recursion relation using renormalized parameters. This procedure ultimately provides information about the infinite lattice. After x iterations, Eq. The renormalized hoppings are smaller than the original t , since they are multiplied by the undressed g , as in Eq.
Those read. After x iterations, we have that site 1 is coupled to a chain of 2 x sites where the effective hopping parameter is much smaller. Thus we have an approximation to the local Green function from the surface site 1, at the edge of the chain:. To have a picture of the decimation procedure, we illustrated the iterations steps in Fig. Note that it is the reverse of the encapsulating mechanism of the infinite lattice into a finite chain, shown in Fig.
We start with the three-site chain, shown in Fig. In the first iteration, we add two interstitial sites, growing the lattice to 5, shown in Fig. With these renormalized parameters one can simulate a chain that grows exponentially fast keeping the three-point structure of Eq. The surface RGF is widely used in transport simulation with several applications [ 44 - 46 ] with sophistications [ 47 ]. In the next section we will present an alternative version, capable to access the Green's functions of the edge and bulk at once, possibly finding usefullness in topological insulators.
The decimation is similar to the surface procedure, we will insert interstitial sites at each iteration. The difference is in which functions we eliminate in the hierarchy of equation of motions and in the recursive model. Although the equation of motion EOM procedure is quite mechanic, we will exemplify how the decimation develops in the first iteration of the surface-bulk RGF.
By now the reader can probably jump into the effective equations, we elaborate them for the sake of clarity. Let us add two sites a and b to the 3-site chain, shown in Fig. For 5 sites, the equations are more numerous and the surface solution will be more intrincate. We will examine three sites, the edges and the central site.
By replacing in , we eliminate the non-diagonal function G a 1. Note that the edge propagator G 11 corresponds to Eq. Eliminating the Green's functions and , we obtain Eq. Finally, the Green's function for the last site of Fig. Comparing these expressions with 97 , we will consider in the renormalization of g 3. In this five-site example we explicited the first step of the decimation recursion based on the three-site system.
This procedure is different from the surface Green's function approach, since we kept the three local propagators, eliminating the non-diagonal ones. Figure 12 illustrates the renormalization of the interactions and the mapping of the five-site chain onto the effective three-site one. Panels c e d illustrate the recursive procedure of encapsulating the new sites to obtain the effective three-site system.
As the decimation procedure is carried, the number of peaks grows with the number of sites. The correspondent source code is presented in the Appendix. The algorithm is shown in the Appendix , source code 2 , which simulates the semiinfinite chain. The surface-bulk RGF decimation technique detailed in Sec. However, both methods scale exponentially with the number of iterations and are easily extended to two-dimensions via a matrix representation. Here we chose to ellaborate better how the proposed surface-bulk decimation works in practice.
We implemented the surface-bulk RGF algorithm in Julia. The source code 2 see Appendix uses the recursive method to evaluate the surface density of states of a semi-infinite linear chain. The results of few steps are plotted in Fig. In order to approach two-dimensional materials, a generalization of the RGF decimation technique is usually performed by slicing a region central device or lead in layers, from which the surface algorithm follows [ 42 ].
In two dimensions it is convenient to adopt a matrix representation of our Green's functions and hoppings. We will approach this generalization in the simplest 2D example of a ladder, where we couple two 3-site chains vertically, as shown in Fig. Each site will be indexed by its column layer i and row j. The new site indexes ij correspond to the column i and row j. Let us consider now displacements both on the horizontal as well as in the vertical direction. For example, the electron in the 11 site can visit the two first neighbors 21 or 12 see Fig.
The equation of motion of the G 11,11 site will exhibit then a self contribution 11 and two non-diagonal propagators G 21,11 e G 12, Notice that Eq.
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Casting the left-hand side l. This is very convenient, since we will be able to implement decimation in two dimensions. Therefore we can also rewrite Eq. From the two identifications above we can perform a mapping to three effective sites, corresponding to these slices, shown in Fig.
The decimation method applies, allowing the simulation e. The program in Julia to generate the results of the ladder is shown in the Appendix. To go beyond the ladder, we can generalize V and W to bigger slices. These matrices will be larger but have a simple form, let us develop them. First note that, in a given slice, the electron can hop up or down a row.
By our definitions see Fig. For the W matrix, the hopping takes place between sites of different columns. Presently we deal with three effective sites, but as the decimation proceeds, the lattice will grow horizontally, forming a stripe. In this process, notice that independently of the column i , automatically all rows j of the slice will be connected since the slices will touch each other. Therefore one can generalize the algorithm of the ladder to a stripe geometry, using the matrices and As we increase the width of the stripe, the behavior tends to the limit of an infinite square lattice, given by an analytic expression in terms an ellyptical function of the first kind [ 48 ].
It is associated with critical saddle points in the two-dimensional band structure [ 49 ]. The analytical result of the infinite square lattice [ 48 ] is shown as a reference of the asymptotic limit. This last example illustrates the power of this technique in simulating finite lattices, which can go beyond the present regular chains to real nano or mesoscopic systems, such as electrodes, cavities, quantum dots and molecular junctions.
To conclude, we have presented a pedagogical introduction que the Green's function in the many-body formalism. Starting with a general view of Green's functions, from the classical mathematical origin, going through the many-body definitions, we finally reached a practical application within the recursive Green's functions technique. For a young researcher, it is not easy to grasp the whole power and at the same time, the tiny details of the numerical methods available. Therefore we prepared this introduction based on simple condensed-matter models with additional implementations in Julia , an open-source high-level language for scientific computing.
The surface-bulk recursive Green's function is, to the best of our knowledge, a new proposal to the field, which brings an advantage in the investigation of topological materials, where one is interested in the edge and the bulk properties.
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Like the surface approach, our surface-bulk recursive Green's function can be generalized to other systems and geometries [ 44 , 47 ]. We believe this material will be also useful for researchers unfamiliar with the Green's function method, interested in the new challenges of nanosciences and their implementations.
We would like to acknowledge Ginetom S. Diniz, Gerson J. Ferreira, and Marcel Novaes for suggestions and careful reading. Julia is a high-level, high-performance, easy-to-learn scientific language [ 50 ]. It is also an open-source project, licensed by MIT. For an introductory course, please see for instance Reference [ 41 ]. In source code 1 , we define a linearly spaced vector of energies using the command linspace and evaluate the undressed Green's function from this vector.
This shortened notation avoids additional and traditional use of the for loop for energies, which is inefficient, since the vector can be stored in memory at once, on the fly. If the amount of data to be stored is under the memory resources, vectorization of loops is a general recommended programming practice, since matrix and vector operations can be performed efficiently in Julia. When we start evaluating more complex Green's functions, stored as large matrices, we return to the conventional loop of energies.
Code 2 uses the recursive method to evaluate the surface density of states of a semi-infinite linear chain. We use again the vectorized loop of energies w in the linspace command. The explicit for loop runs the recursive decimation procedure for 16 steps. Equations , and are implemented inside the loop. Next we renormalize the hoppings and the undressed Green's functions, carrying the decimation.
In the last lines we plot the local density of states of site 1, the local Green's function is given by Eq. In source code 3 , we have implemented the decimation using the matrix forms in Julia. We had to define a vertical and horizontal hopping parameters, tv and tw , along with hopping matrices V and W. We now perform an explicit energy and decimation loops, iterating for energy points and 18 decimation steps.
Before decimating, we construct a pair of sites, described by the dressed function gV , Eq. As shown in Fig. The decimation loop is the same of source code 2 , except for the fact that we have now a hopping matrix W. After the loop, we evaluate the three local functions as in Eq. Cannell and S. Krantz, The Mathematical Intelligencer 26 , 68 Cannell and N.
Lord, The Mathematical Gazette 77 , 26 Green, arXiv preprint arXiv D Cole, J. Beck, A. Haji-Sheikh and B. Amrein, A. Hinz and D. Taketani and M. Wheeler and W. Lippmann and J. Schwinger, Physical Review 79 , Sakurai and J. Bruus and K.
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Dyson, Physical Review 75 , Dirac, Physikalische Zeitschrift der Sowjetunion 3 , 64 Feynman, Review of Modern Physics 20 , Feynman, Physical Review 76 , Lancaster and S. Martin and J. Schwinger, Physical Review , Baym and L. Kadanoff, Physical Review , Kadanoff and G. Altland and B. Zubarev, Soviet Physics Uspekhi 3 , Haug and A. Cuevas and E.
Stefanucci and R. Haydock, Solid State Physics, vol. Ferreira, Introduction to Computational Physics with examples in Julia unpublished, Guinea, C. Tejedor, F. Flores and E. Louis, Physical Review B 28 , Lopez Sancho, J. Sancho and J. Lewenkopf and E.
Mucciolo, Journal of Computational Electronics 12 , Nardelli, Physical Review B 60 , Pauly, J. Viljas, U. Huniar, M. Wohlthat, M. Cuevas and G. Thorgilsson, G. Viktorsson and S. Erlingsson, Journal of Computational Physics , Bezanson, A. Edelman, S. Karpinski and V. Shah,arXiv preprint arXiv Today, Green's identities are a set of three vector equations relating the bulk with the boundary of a region on which differential operators act, closely related to Gauss' divergence and Stokes' curl theorems. Green's second identity allows the conversion of a triple integral of laplacians within a volume into a double integral of gradients over its surface boundary:.
Later we will return to this point. This means that the potential V in the Hamiltonian will be present e. In practice, the solvable system will be a building block for the more complex ones. Similarly, in the presence of spacial translational symmetry, the representation in the momentum space is also convenient. We thank K. Pototzky for this remark. Thus in this problem spatial degrees of freedom are decoupled from spin, since nor the kinetic energy nor the local potential couples to the spin of the particles.
For an analytic function in the upper plane, the Hilbert transform describes the relationship between the real part and the imaginary part of the boundary values. This means that these functions are conjugate pairs. One can consider an additional site and couple it from the left and from the right with semi-infinite chains, as we have shown in Fig. To this, one should first determine the surface GF from both sides, which usually are identical.
However, they can differ for instance in topological systems, where each side has its own chirality, or for asymmetric leads in transport devices.
License information: This is an open-access article distributed under the terms of the Creative Commons Attribution License type CC-BY , which permits unrestricted use, distribution and reproduction in any medium, provided the original article is properly cited. Services on Demand Journal. Articles Pedagogical introduction to equilibrium Green's functions: condensed-matter examples with numerical implementations. Vernek 1. Introduction The Green's functions method is a powerful mathematical tool to solve linear differential equations. Classical Green's functions Formally, a Green's function is a solution of a linear differential equation with a Dirac delta inhomogeneous source sometimes referred as a delta or unit pulse with homogeneous boundary conditions.
For example, in an ordinary differential equation, it should read: 2. Quantum Green's functions By the beginning of the 20th century, Green's functions were generalized to the theory of linear operators, in particular, they were applied to the class of Sturm-Liouville operators [ 14 ]. Electron Green's function We will start with formal definitions of the electron Green's function, our object of study. For instance, the so-called "causal" Green's function reads Spectral representation So far we have presented the Green's function in the time domain. To illustrate this, let us first consider the spectral representation in the special case of a free particle Hamiltonian, which can be written as The equation of motion technique One way of obtaining the Green's function is to determine its time evolution via equation of motion EOM technique.
Simple example: the non-interacting linear chain Let us consider a linear chain described by the non-interacting Hamiltonian containing a single orbital energy per site and a kinetic term that connects all nearest-neighbor sites via a hopping parameter t Two-site chain: the hydrogen molecule The simplest finite lattice has only two sites, see Fig. Semi-infinite linear chain An interesting example that provides an analytical closed solution of the equations of motion is the semi-infinite linear chain, shown in Fig.
Infinite linear chain Another interesting model that allows analytical solution is the infinite linear chain. Three-site chain: a recipe for recursion Let us now apply the equation-of-motion technique to a linear chain composed of three sites, shown in Fig. Recursive Green's function 3. Surface Green's functions decimation In early 80's, the investigation of surface and bulk properties of metals, transition metals and semiconductors motivated the development of effective Hamiltonians and iterative techniques to obtain the density of states [ 42 ].
Again, for the first site we have the equations of motion The ladder In order to approach two-dimensional materials, a generalization of the RGF decimation technique is usually performed by slicing a region central device or lead in layers, from which the surface algorithm follows [ 42 ].
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Conclusions To conclude, we have presented a pedagogical introduction que the Green's function in the many-body formalism. Source codes Julia is a high-level, high-performance, easy-to-learn scientific language [ 50 ]. References  D. Green's second identity allows the conversion of a triple integral of laplacians within a volume into a double integral of gradients over its surface boundary: 1.
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