CITE THIS NOTEBOOK: Hyperbolic matter in electrical circuits with tunable complex phases by Anffany Chen. Wolfram Community FEB 23 2023.
ORIGINAL ARTICLE: Chen, A., Brand, H., Helbig, T. et al. Hyperbolic matter in electrical circuits with tunable complex phases. Nature Communications 14, 622 (2023). https://doi.org/10.1038/s41467-023-36359-6.
Article Abstract: Curved spaces play a fundamental role in many areas of modern physics, from cosmological length scales to subatomic structures related to quantum information and quantum gravity. In tabletop experiments, negatively curved spaces can be simulated with hyperbolic lattices. Here we introduce and experimentally realize hyperbolic matter as a paradigm for topological states through topolectrical circuit networks relying on a complex-phase circuit element. The experiment is based on hyperbolic band theory that we confirm here in an unprecedented numerical survey of finite hyperbolic lattices. We implement hyperbolic graphene as an example of topologically nontrivial hyperbolic matter. Our work sets the stage to realize more complex forms of hyperbolic matter to challenge our established theories of physics in curved space, while the tunable complex-phase element developed here can be a key ingredient for future experimental simulation of various Hamiltonians with topological ground states.
Introduction
Hyperbolic lattices are two-dimensional lattices with constant negative spatial curvature. The simplest hyperbolic lattices are formed by p-sided polygons with q of them meeting at each vertex (denoted {p, q} by the Schläfli symbol). The adjacency matrix of {p, q} lattice, when multiplied by hopping amplitude -J, gives rise to a tight-binding Hamiltonian describing the energies and dynamics of non-interacting particles traveling on the lattice. In contrast to flat Euclidean lattices, the boundary of a finite-sized hyperbolic lattice (with open boundary condition) constitutes a significant portion of the total system regardless of the system size, so the boundary region is never negligible. This Notebook demonstrates two approaches to realize hyperbolic lattices without boundary, suitable for numerical studies of hyperbolic bulk properties. In Chapter 1, we utilize hyperbolic band theory to generate the momentum-space Hamiltonian of lattice {8, 3}. In Chapter 2, we import and numerically diagonalize the adjacency matrix of a {8, 3} regular map, which is a finite {8, 3} hyperbolic lattice with periodic boundary condition.
Chapter 1: Bloch Hamiltonian by Hyperbolic Band Theory
The Bloch Hamiltonian of lattice {8, 3} is generated below . One can multiply the resulting matrix by an overall factor of hopping amplitude -J . For numerical diagonalization, substitute the momentum components
k
μ
by numbers .

{8, 3} Lattice

In[]:=
Clear["Global`*"]

Define useful functions and quantities

In[]:=
p=8;q=3;pB=8;qB=8;​​Nc=16;(*sizeofunit-cell*)
In[]:=
dist[z_,z2_]:=ArcCosh1+
2
2
Abs[z-z2]
(1-
2
Abs[z]
)(1-
2
Abs[z2]
)
;(*hyperbolicdistance*)​​Act[Op_,Site_]:=(Op[[1,1]]Site+Op[[1,2]])/(Op[[2,1]]Site+Op[[2,2]])(*actionofOperatoronSite*)​​R[ϕ_]:={{Exp[ϕ/2],0},{0,Exp[-ϕ/2]}}(*rotationmatrix*)
In[]:=
r0=
Cos
π
p
+
π
q

Cos
π
p
-
π
q

;​​d0=dist[r0,r0Exp[2π/p]];(*distancebetweennearestneighbours*)

Coordinates of unit-cell sites

In[]:=
a=Cosh[dist[r0,0]+d0]-1;r=Sqrt[a/(2+a)];​​UnitCell=Join[Table[r0Exp[(2πn/p+π/p)],{n,0,p-1}],Table[rExp[(2πn/p+π/p)],{n,0,p-1}]]//N;

Fuchsian group generators, including inverses

In[]:=
α=2π/pB;β=2π/qB;σ=Sqrt[(Cos[α]+Cos[β])/(1+Cos[β])];​​γ0={{1,σ},{σ,1}}/Sqrt[1-σ^2];​​G1=Table[R[nα].γ0.R[-nα],{n,0,pB}];

Products of 2 or 3 generators, with all possible permutations

In[]:=
G2=Table[G1[[n]].G1[[m]],{n,pB},{m,pB}];​​G3=Table[G1[[n]].G1[[m]].G1[[s]],{n,pB},{m,pB},{s,pB}];

Construct the Bloch Hamiltonian by searching for the nearest neighbours of each site in the unit-cell


Reduce the number of momentum components to 2g

In[]:=
HB=FullSimplify[H/.{
k
5
-
k
1
,
k
6
-
k
2
,
k
7
-
k
3
,
k
8
-
k
4
}];
In[]:=
HB//MatrixForm
Out[]//MatrixForm=
0.
1.
0.
0.
0.
0.
0.
1.
1.
0.
0.
0.
0.
0.
0.
0.
1.
0.
1.
0.
0.
0.
0.
0.
0.
1.
0.
0.
0.
0.
0.
0.
0.
1.
0.
1.
0.
0.
0.
0.
0.
0.
1.
0.
0.
0.
0.
0.
0.
0.
1.
0.
1.
0.
0.
0.
0.
0.
0.
1.
0.
0.
0.
0.
0.
0.
0.
1.
0.
1.
0.
0.
0.
0.
0.
0.
1.
0.
0.
0.
0.
0.
0.
0.
1.
0.
1.
0.
0.
0.
0.
0.
0.
1.
0.
0.
0.
0.
0.
0.
0.
1.
0.
1.
0.
0.
0.
0.
0.
0.
1.
0.
1.
0.
0.
0.
0.
0.
1.
0.
0.
0.
0.
0.
0.
0.
0.
1.
1.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

k
1

0.

k
2

0.
0.
0.
1.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

k
2

0.

k
3

0.
0.
0.
1.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.

k
3

0.

k
4

0.
0.
0.
1.
0.
0.
0.
0.
-
k
1

0.
0.
0.
0.
0.

k
4

0.
0.
0.
0.
0.
1.
0.
0.
0.
0.
-
k
2

0.
0.
0.
0.
0.
-
k
1

0.
0.
0.
0.
0.
1.
0.
0.
-
k
2

0.
-
k
3

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
1.
0.
0.
-
k
3

0.
-
k
4

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
1.
0.
0.
-
k
4

0.

k
1

0.
0.
0.
Chapter 2: Regular Map
The graph data for the {8, 3} regular map below (attached at the end of this post) is labeled by the Conder index and obtained from the following reference:
Conder, M., "Trivalent (cubic) symmetric graphs on up to 2048 vertices", (2006),
​https://www.math.auckland.ac.nz/~conder/symmcubic2048list.txt [Online; Accessed: 2022-02-07]

{8, 3} C2048.23

In[]:=
Clear["Global`*"]
In[]:=
SetDirectory[NotebookDirectory[]];
In[]:=
ImpFile=StringSplit[Import["C2048.23.graph.txt"]];​​ImpFile2=Table[ImpFile[[i]],{i,4,Length[ImpFile]}];​​Neighbors=ToExpression[Flatten[Table[{ImpFile2[[i]],ImpFile2[[i+1]],ImpFile2[[i+2]],ImpFile2[[i+3]]},{i,1,Length[ImpFile2],5}]]];​​LengthA=
1
4
Length[Neighbors];​​Amat=Table[0,{i,1,LengthA},{j,1,LengthA}];​​Ji=1;Do[​​Amat[[Neighbors[[Ji]],Neighbors[[Ji+1]]]]=1;​​Amat[[Neighbors[[Ji]],Neighbors[[Ji+2]]]]=1;​​Amat[[Neighbors[[Ji]],Neighbors[[Ji+3]]]]=1;​​Ji=Ji+4,{LengthA}];​​Total[Amat]//Tally​​Spec=Eigenvalues[-Amat//N]//Sort;
Out[]=
{{3,2048}}
In[]:=
ListPlot[Spec]
Out[]=
500
1000
1500
2000
-3
-2
-1
1
2
3
In[]:=
Short[Spec,10]
Out[]//Short=
{-3.,-2.84776,-2.84776,-2.84776,-2.84776,-2.84776,-2.84776,-2.84776,-2.84776,-2.84776,-2.84776,-2.84776,-2.84776,-2.80513,-2.80513,-2.80513,-2.80513,-2.80513,-2.80513,-2.80513,-2.80513,2006,2.80513,2.80513,2.80513,2.80513,2.80513,2.80513,2.80513,2.80513,2.84776,2.84776,2.84776,2.84776,2.84776,2.84776,2.84776,2.84776,2.84776,2.84776,2.84776,2.84776,3.}