Complete Solution of the Tight Binding Model on a Cayley Tree

The complete set of Eigenstates and Eigenvalues of the nearest neighbour tight binding model on a Cayley tree with branching number $b=2$ and $M$ branching generations with open boundary conditions is derived. These results are used to derive the total and the local density of states.


I. INTRODUCTION
Arthur Cayley introduced the Cayley tree graph as a graphical representation of the free group 1 . The Cayley tree is a tree graph with N nodes, branching number b with degree k = b + 1, except at surface edge nodes where k = 1. Since it is loop free, the dynamics on Cayley trees is amenable to exact solutions employing the transfer matrix method. The local density of states at the central site of a tight binding model on a Cayley tree has been derived analytically in Refs. [2][3][4][5] . The tight binding model for disordered fermions has been solved analytically by the transfer matrix method on a Cayley tree, revealing the Anderson delocalization transition for b > 1 [6][7][8][9] . For infinite number of lattice sites the Caylee tree is called Bethe lattice since Bethe' s approximation for the Ising model becomes exact on this lattice 10 . Other interacting models, in particular the Hubbard model have been studied on the Bethe lattice. Since in the limit of k → ∞ mean field theory for any model with interactions becomes exact, the formulation on the Bethe lattice has been used to study this limit in a controlled way 11 . The problem of quasiparticle relaxation in an interacting electron system has been mapped on the localization problem in Fock space and solved approximately by mapping it on a Cayley tree 12 . Recently, the dynamics of coupled oscillators have been studied on a Cayley tree, as a model for the dynamics in distribution power grids 13 .
Inspite of this wide range of applications of the Cayley tree in physics, the Eigenstates and Energy Eigenvalues of the tight binding model have hardly been studied. In 2001, Mahan obtained the shell symmetric Eigenstates on a Bethe lattice and derived from it the local density of states at the central site 4 . However, the full basis of Eigenstates on a Cayley tree was not obtained there. We therefore intend to fill this gap in this paper for branching number b = 2.

II. THE TIGHT BINDING MODEL ON A CAYLEY TREE
The tight binding model is defined bŷ where t ij is the hopping amplitude between sites i and j on the lattice of N sites. < i, j > denotes nearest neighbours on the graph. We will assume homogenous hopping amplitude t ij = t, in the following. We are interested in obtaining the full set of Eigenstates |Ψ n with Eigenvalues E n as given bŷ for all n = 1, ..., N . Here, we consider the sites to be on a Cayley tree of branching number b = 2, as shown in

A. Choice of basis
Mahan found a subset of M + 1 Eigenstates of all N Eigenstates on a Cayley tree 4 by using shell symmetric states as a subbasis for the Eigenstates. These shell symmetric basis states are symmetric with respect to a rotation between different branches of the Cayley tree, which are highlighted by different colors in Fig. 1. The Eigenstates have therefore equal amplitude on all sites of the same generation l. Accordingly, we can label these symmetric states with one number that denotes the generation l of the hopping starting from the central site, e.g. |l = 3 denotes the symmetric state on all 3 rd generation sites. Eq. (2) then furnishes recurrence relations. These were solved by Mahan to obtain solutions with equal amplitude on sites of same generation (symmetric solutions) 4 .
In order to obtain all Eigenstates, we need to extend the basis to all states to be able to distinguish between the different branches of the Cayley tree. In a first step, let us split the tree into three main branches starting at the central site as shown in Fig. 1. We denote with |l m the normalized symmetric combination of local states defined on the nodes of the l th generation in branch m, where l = 0 denotes the central node of the Cayley tree. We enumerate the three branches originating from the l = 0 site with m ∈ {1, 2, 3}.
For example in Fig. 1 Thereby we get 3M +1 basis states. In order to get the remaining basis states, we include successively all antisymmetric superpositions of site states which branch from a node α in the l th generation of the Cayley tree to the right and left as shown in Fig. 2. The basis states are taken then to be the antisymmetric combination of all site states of the l + r-generation in the left and right branches evolving from node α. Since there are 3 × 2 l−1 states in each generation l, we enumerate these states with α ∈ {1, 2, ..., 3 × 2 l−1 }. Such a state starting in generation l is thus denoted as |l, r α with r ∈ {1, .., M − l}.
Thus, for each node α in the l th generation we get M − l such child states |l, r α , with r ∈ {1, .., M − l}. Since there are 3 × 2 l−1 nodes α in the l th generation, the total number of such states is, Thus, this completes the N basis states forming an orthonormal basis for the Hilbert Space of the tight binding model on the Cayley tree with N sites. Using this basis simplifies the solution of the eigenvalue equation (2), since it can be arranged in blocks, as we will see in the following section.

B. Block Recurrence Relations
Any eigenstate |Ψ can now be written as a superposition of the basis states where ψ 0 , φ α l,r and ψ l,m are complex amplitudes and G l denotes the set of all 3 × 2 l−1 sites in the l th generation of the Cayley tree.

C. Solutions of the recurrence relations
Let us start with solutions which satisfy ψ 0 = 0, with which Eqs. 7 yield, and for l = 1, ..., M. This gives, for l = 1, ..., M − 1. With the Ansatz for the Energy Eigenvalues E = 2 √ 2 cos θ, we get that For now, ψ 1,m can be freely choosen provided Eq. 9 is satisfied. This will be fixed later on by requiring that the wavefunction be normalized. The Eqs. 7 are closed by the open boundary condition at the surface of the Cayley tree, We note that ψ 1,m = 0 for at least one m ∈ {1, 2, 3}. Eq. (13) requires together with Eq. (11) the quantisation condition Thus, we get the following discrete solutions for θ with i ∈ {1, 2, ..., M }. This gives the discrete energy eigenvalues Next, we get ψ 1,m by imposing the normalization condition which with Eq. (11) gives, We can eliminate ψ 1,3 with Eq. (9) to get, Defining ψ 1,1 := r 1 e iν1 and ψ 1,2 := r 1 e iν2 we get which, for fixed energy E, gives a parameter family of ellipses for different ∆ν := ν 1 − ν 2 . For fixed energy E we find the two orthogonal solutions to Eq. 19 for arbitrary real phases µ and δ. Thus, all other solutions of Eq. 19 are linear combinations of these solutions. Next, using Eq. 11 and Eq. 9, we get all remaining complex amplitudes ψ l,m . Thus, for each possible energy eigenvalue E i , given by Eq. (16), we get two degenerate orthogonal eigenstates with the following amplitudes on the basis vector components where the two possible choices of ψ 1,1 and ψ 1,2 , as given by Eqs. (21), give two orthogonal eigenstates with the same energy E i . Since there are M possible values of E i , Eq. (16) and Eq. (15) and each Eigenspace is two fold degenerate, the total number of states of this kind is 2M .
The Eigenstates given by Eq. 22 are orthogonal to Mahan's symmetric solutions, since the basis states in Mahan's analysis have equal weight in all three branches, and thus ψ l,1 = ψ l,2 = ψ l,3 for all l. Since there are M + 1 Mahan's solutions and they are orthogonal to the 2M solutions obtained above in Eq. 22, we have obtained all the solutions we can get from the subset of 3M + 1 basis states obtained from Eq. 7. Mahan's solutions are, in our notation, given by The solutions to the recursive equations obtained from the Eigenvalue equation Eq. 2 with E = 2 √ 2 cos θ are given as 4 : where C is a normalization constant and the phases β and γ are related to θ by the equations,  Having solved the first set of recursion equations Eq. (7), finding 3M + 1 Eigenstates, we move on to solve Eqs. (8) in order to find the remaining Eigenstates. For given integer l and α ∈ G l , the second block of equations in Eqs. (8) resembles the set of equations one obtains for the Eigenstates of a tight binding Hamliltonian on an one-dimensional chain with M − l sites. Thereby, we can readily write its solutions as where r = 2, ..., M − l − 1, with energy eigenvalues The possible values of χ, for each choice of l and α, is obtained from the open boundary conditions φ α M +1,r = 0, yielding the quantisation condition In particular, we obtain the surface states for l = M − 1, where we get Eq. (31) dictates that χ = π 2 , and thus the Eigen energy of the surface states is E = 0 with eigenstates given by where n = 2 √ 2 cos θ n . The summation can be approximated by an integral over θ in the limit of a Bethe lattice, M → ∞, where one finds 4 ρ 00 (E) = π 0 dθ 2 π sin 2 (β(θ))δ(E − n ) = 3 2π Having obtained all the eigenstates and energies of the Schroedinger equation on the Cayley tree, we can now proceed to calculate the total density of states, given by where the sum is over all Eigen energies n . It is convenient to write it as a sum of contributions from the threee different different kind of states I, II, III, which we found above, where ρ I denotes the contribution due to the M +1 symmetric Mahan states, ρ II the contribution due to the 2M states which have same amplitude in each of the three branches and ρ III the contribution due to the N − (3M + 1) states which are localised in different branches of the Cayley tree. In the large M limit, we get see the Appendix for the derivation. Similarly, we find and where D l = α∈G l = 3×2 l−1 is the number of sites in the l th generation. We observe that the degeneracy of the states increases with M as ∼ 2 M for the III−type Eigenstates which are localized near the surface. Thus, in the M → ∞ limit, those states are highly degenerate.
As an example we show the results for a numerical computation for a Cayley tree with M = 3 generations. We see the distribution of energy eigenvalues and their degeneracies in Fig. 3, in particular that the states at zero energy E = 0 are highly degenerate. The histogram is symmetric about E = 0, with same number of states with energy −E as with energy E as is clear from the analytical solution Eqs. 15, 28, 31 for M = 3.

V. CONCLUSION
The complete set of Eigenstates and Eigenvalues of the nearest neighbour tight binding model on a Cayley tree with branching number b = 2 and M branching generations with open boundary conditions has been derived. Besides the M + 1 shell symmetric Eigenstates derived already by Mahan in Ref. 4, we find 2M Eigenstates which have zero amplitude at central site but are otherwise extended throughout the Cayley tree. The remaining N − (3M + 1) states are found to be strongly localised states with finite amplitudes on only a subset of sites. In particular, there are 3 × 2 M −2 states which are each antisymmetric combinations of two sites at the surface of the Cayley tree and exact Eigen energy E = 0. These results are used to derive the total density of states as function of energy E. Having all Eigenstates and Eigenfunctions enables one to derive the local density of states not only at the central site as in Ref. 4 but at any site. Moreover, it allows one to derive any one particle observable on the Cayley tree, as well as one particle response functions. Using similar strategies for choosing a convenient basis, the Eigenstates for any branching number b > 2 can be derived, a task we leave for a future publication. .