Spectra of Quantum Graphs

Transcription

1 Spectra of Quantum Graphs Alan Talmage Washington University in St. Louis October 8, 2014 Alan Talmage (2014) Spectra of Quantum Graphs October 8, / 14

2 Introduction to Quantum Graphs What is a Quantum Graph? A metric graph is a graph in which each edge has a length L (0, ). A quantum graph is a metric graph together with the Schrödinger Hamiltonian d2 dx 2 + V (x) and an appropriate vertex condition. Alan Talmage (2014) Spectra of Quantum Graphs October 8, / 14

3 Introduction to Quantum Graphs Vertex Conditions We will be using the Neumann boundary conditions at each vertex v: 1 f (x) is continuous at v 2 e E(v) f e (v) = 0, i.e. the flux of f through the vertex v is zero. These are the boundary conditions for electrical networks, so using these boundary conditions allows us to model a particle on an electrical network. The motivation for the study of quantum graphs comes primarily from the study of crystal lattices. By treating atoms as vertices and bonds as edges, we can approximate a crystal lattice as a quantum graph. The spectrum of the Schrödinger Hamiltonian will then give the allowable energy states of the lattice. Alan Talmage (2014) Spectra of Quantum Graphs October 8, / 14

4 Graphene Graphene Lattice Graphene is a two-dimensional hexagonal lattice of carbon atoms. It is an obvious choice for this type of analysis, since it is a substance whose unique properties derive specifically from its geometry. Alan Talmage (2014) Spectra of Quantum Graphs October 8, / 14

5 Graphene Assumptions To make this a quantum graph, we make the following assumptions: 1 Every edge has length 1 (and we will thus identify it with the interval [0,1] ). 2 The potential V (x) of our Hamiltonian H = d 2 dx + V (x) is the same 2 on each edge, and V (x) = V (1 x) on an edge [0,1]. 3 We are using the Neumann boundary conditions. Alan Talmage (2014) Spectra of Quantum Graphs October 8, / 14

6 Graphene Fundamental Domain To study the hexagonal lattice, we begin by choosing this fundamental domain. The rest of the lattice can be obtained from this domain by translations. Alan Talmage (2014) Spectra of Quantum Graphs October 8, / 14

7 Graphene Floquet-Bloch Analysis We are interested in studying the effect the geometry of the graph has on the spectrum σ(h), so we want to split the analysis into analysis on a single edge, which will be the same for every graph, and analysis on the graph structure. We define the Bloch Hamiltonian H θ for θ = (θ 1, θ 2 ) [ π, π] 2, which acts in the same way as H on the fundamental domain, but with the cyclic boundary conditions u(x + p 1 e 1 + p 2 e 2 ) = e i(p 1θ 1 +e 2 θ 2 ) u(x), where e 1, e 2 are basis vectors for the translational symmetries of the graph.. Theorem We can then apply the following result from Floquet theory: λ σ(h) λ σ(h θ ) for some θ [ π, π] 2 Alan Talmage (2014) Spectra of Quantum Graphs October 8, / 14

8 Graphene Spectrum of the Graphene Lattice This means that the dispersion relation θ σ(h θ ) contains the information about the spectrum that is specific to the graph structure. So we can restrict our analysis to a single fundamental domain of the graph. The computation of the dispersion relation is fairly straightforward at this point: I will omit the computations. The dispersion relation for the hexagonal lattice is: ± cos θ 1 θ 2 cos θ cos θ 2 2 Note that this relation has two branches. Alan Talmage (2014) Spectra of Quantum Graphs October 8, / 14

9 Other Graphs Square Lattice We can apply the same methods to the square lattice, using this fundamental domain: The dispersion relation is cos(θ 1 ) + cos(θ 2 ) Alan Talmage (2014) Spectra of Quantum Graphs October 8, / 14

10 Other Graphs Spectrum of Z n This technique is not limited to two dimensions: it works on any graph that can be formed from a fundamental domain translated by some Z n. For example, the dispersion relation for the n-dimensional cubical lattice is 2 n cos(θ i ) n 1 i=1 Alan Talmage (2014) Spectra of Quantum Graphs October 8, / 14

11 Other Graphs Ladder Graph Another interesting graph whose spectrum is easy to compute is this ladder graph: Its spectrum has two branches: 2 (2 ± 1)cos(θ) 3 Note that the outer branch is identical to the dispersion relation of the 1-dimensional cubical lattice, and the inner branch is the same, but scaled down by a constant. Alan Talmage (2014) Spectra of Quantum Graphs October 8, / 14

12 Possible Algebraic Connections Cayley Graphs The fact that the spectrum of the ladder graph is very similar to the spectrum of the line, as well as the fact that adding dimensions to the cubical lattice only adds another copy of the spectrum of the line graph (up to normalization) suggests that there may be an underlying algrebraic structure. One connection between algebraic structure and graph structure we can study is the relation between groups and their Cayley graphs. For example, the n-dimensional cubical lattice is the Cayley graph of Z n, and the ladder graph discussed above is the Cayley graph of Z 2 Z. Alan Talmage (2014) Spectra of Quantum Graphs October 8, / 14

13 Possible Algebraic Connections Further Research Directions While there are clearly some similarities between the algebraic structure of a group and the quantum graph structure of its Cayley graph, the exact relationship has not been characterized. The method employed here is not applicable to the Cayley graph of a nonabelian group. Not all graphs are Cayley graphs of some group. For example, the hexagonal graphene lattice, which provided the original motivation for studying this topic, is not. Thus, other methods will be needed to characterize their structure. Alan Talmage (2014) Spectra of Quantum Graphs October 8, / 14

14 Acknowledgements Acknowledgements This research was performed under the direction of Professor Renato Feres, whose guidance was invaluable. The computation of the spectrum for the graphene lattice is from Kuchment, P. and Post, O. On the Spectra of Carbon Nano-structures ArXiv, 2008 This material is based upon work supported by the National Science Foundation under agreement No. DMS Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Alan Talmage (2014) Spectra of Quantum Graphs October 8, / 14