6
$\begingroup$

I have a lattice described by a AdjacencyMatrix or Hamiltonian. This matrix has all the information of how each site is connected to another site with what hopping amplitude. For present case, all the hopping amplitudes are 1.
What I want to do:

  1. Plotting of probability density $|\Psi_{i,j}|^2$ on this lattice, where $\{i,j\}$ are the lattice sites.
  2. Dynamics: On the same lattice, I would be interested if I excite a lattice site at a position $\{i_0,j_0\}$ (initial condition), then how will that excitation or wave propagates in time. Essentially, I am interested in solving the Schrodinger's wave equation. Then looking for the wave solution. For instance, if we excite an eigenstate, then this initial state will remain so even after a long lapse of time. This dynamics is to be seen on the lattice at different discrete times.

My MWE:

(* The Hamiltonian or AdjacencyMatrix of the lattice. Here it is a \
hexagonal lattice *)

hmat = need to be copied from this link

(* The Adjacency graph looks like: hexagonal lattice. *)
  AdjacencyGraph[hmat]

(* Putting coordinates on the hexagonal lattice. *)
  imax = 4; (* the number of rings in the lattice *)

  coords = 
 ArrayFlatten[
  ParallelTable[Table[{i, j}, {j, 12 (i - 1) + 6}], {i, imax}], 1]

Just to illustrate the above line i.e. coords. The coordinates of the lattice look like (below figure), where the first two rings are highlighted with thick black and red color with proper coordinates going radially. In total, there are four rings imax=4, which can increase or decrease. enter image description here.

(* Plotting of probability density $|\Psi_{i,j}|^2$ on this lattice *)
vecs = Eigenvectors[hmat // N];

Manipulate[
 ListDensityPlot[
  Table[Append[coords[[i]], Abs[vecs[[n, i]]]^2] // N, {i, 
    Length[vecs[[n]]]}]], {n, 1, Length[vecs], 1}]

However, the ListDensityPlot comes in a non friendly way, i.e. not in the form of original lattice. Here, x axis is i label and y axis is the j label. It would have been great if it comes in the radial geometry $(r, \theta$). This I can't seem to resolve. enter image description here

(* The dynamics by solving the Schrodinger equation *)
V[t_] := Table[v[i][t], {i, Length[vecs]}];
sol = NDSolve[{I V'[t] == hmat . V[t], 
    V[0] == Table[KroneckerDelta[i, 10], {i, Length[vecs]}]}, 
   V[t], {t, 0, 100}];

(* The dynamics plot is generated here, but again the plot comes in the same fashion as above not in the usual lattice as shown in the Adjacency graph *)
Manipulate[
 ListDensityPlot[
  Table[N[Append[coords[[i]], Abs[Flatten[V[t] /. sol][[i]]]^2]], {i, 
     Length[vecs]}] /. t -> t1, PlotRange -> {0, 0.05}], {t1, 0, 10}]

Is there a way to resolve this?

$\endgroup$

2 Answers 2

10
$\begingroup$

Try this:

g = AdjacencyGraph[hmat];
graphCoords = GraphEmbedding[g];
Manipulate[
 ListDensityPlot[
  Table[Append[graphCoords[[i]], Abs[vecs[[n, i]]]^2] // N, {i, 
    Length[vecs[[n]]]}]], {n, 1, Length[vecs], 1}]

Hexagonal lattice density plot

$\endgroup$
2
  • $\begingroup$ Thanks a lot for this beautiful answer! Is there a way to show the edges and vertices in the DensityPlot? $\endgroup$
    – Shamina
    Jul 21, 2021 at 10:44
  • 1
    $\begingroup$ You mean like this: Show[ListDensityPlot[...], g]? $\endgroup$
    – Domen
    Jul 21, 2021 at 12:29
6
$\begingroup$

There are nice pictures if we use for visualization Hue and Table

g = AdjacencyGraph[hmat];
imax = 4;
 vecs = Eigenvectors[hmat // N];
graphCoords = GraphEmbedding[g];
Table[ListDensityPlot[
  Table[Append[graphCoords[[i]], Abs[vecs[[n, i]]]^2] // N, {i, 
    Length[vecs[[n]]]}], ColorFunction -> Hue, PlotRange -> All, 
  Frame -> False, PlotLabel -> n], {n, 1, Length[vecs], 1}]

Figure 1 Also for dynamics it is better to use animation like this

frames = Table[
   ListDensityPlot[
    Table[N[Append[graphCoords[[i]], 
        Abs[Flatten[V[t] /. sol][[i]]]^2]], {i, Length[vecs]}] /. 
     t -> t1, PlotRange -> All, ColorFunction -> Hue, Frame -> False, 
    ImageSize -> Small, PlotLabel -> t1], {t1, 0, 10, .2}];
ListAnimate[frames]

Figure 2

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.