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Energy functions of general dimensional diamond crystals based on the Kitaev model

Akito Tatekawa

TL;DR

This work generalizes the Kitaev spin system to the $d$-dimensional diamond lattice $Δ_d$ by building Majorana operators from Clifford algebra irreducibles and formulating a Hamiltonian on a base graph $X_0$. Through a Fourier transform based on the Sunada–Kato–Richard framework, it derives the energy functions $ξ(q)=±2|J_1+∑_{i=1}^{d} J_{i+1} e^{i q·α_i}|$ that exhibit a two-band structure and, crucially, shows these spectra coincide with those of an associated tight-binding model upon identifying $t_i=2J_i$. The paper also analyzes when zeros and energy gaps occur by polygon-type inequalities on the couplings and maps the Kitaev spectrum to Bloch form, highlighting the spectral equivalence and its dependence on the parameter region $Ω_d$. Overall, the results extend exactly solvable spectral insights from the honeycomb case to general dimensional diamond lattices, with clear connections to conventional tight-binding physics and potential implications for band topology and gap engineering.

Abstract

The purpose of this paper is to extend the Kitaev model to a general dimensional diamond crystal. We define the Hamiltonian by using representations of Clifford algebras. Then we compute the energy functions. We show that the energy functions are identified with those appearing in the tight binding model.

Energy functions of general dimensional diamond crystals based on the Kitaev model

TL;DR

This work generalizes the Kitaev spin system to the -dimensional diamond lattice by building Majorana operators from Clifford algebra irreducibles and formulating a Hamiltonian on a base graph . Through a Fourier transform based on the Sunada–Kato–Richard framework, it derives the energy functions that exhibit a two-band structure and, crucially, shows these spectra coincide with those of an associated tight-binding model upon identifying . The paper also analyzes when zeros and energy gaps occur by polygon-type inequalities on the couplings and maps the Kitaev spectrum to Bloch form, highlighting the spectral equivalence and its dependence on the parameter region . Overall, the results extend exactly solvable spectral insights from the honeycomb case to general dimensional diamond lattices, with clear connections to conventional tight-binding physics and potential implications for band topology and gap engineering.

Abstract

The purpose of this paper is to extend the Kitaev model to a general dimensional diamond crystal. We define the Hamiltonian by using representations of Clifford algebras. Then we compute the energy functions. We show that the energy functions are identified with those appearing in the tight binding model.

Paper Structure

This paper contains 9 sections, 6 theorems, 80 equations, 8 figures.

Table of Contents

  1. Introduction
  2. d-dimensional diamond crystal $\Delta_d$
  3. The Kitaev model
  4. Representations of Clifford algebras and $\Delta_d$
  5. The Hamiltonian of of the Kitaev model for $\Delta_d$
  6. Fourier transform
  7. Zeros of the energy functions and energy gaps.
  8. Relation with the tight binding model
  9. Acknowledgments

Key Result

Lemma 4.1

The points $P_1$ and $P_2$ belong to the interior of $D'_{\beta_i}$ and there are no other vertices of $V(\Delta_d)$ belonging to $D'_{\beta_i}$.

Figures (8)

  • Figure 1: The honeycomb lattice $\Delta_2$ and the diamond crystal $\Delta_3$
  • Figure 2: Links of the honeycomb lattice
  • Figure 3: Majorana operators of the honeycomb lattice
  • Figure 4: Minimum ground state energy functions of the honeycomb lattice
  • Figure 5: The base graph of $d$-dimensional diamond crystal
  • ...and 3 more figures

Theorems & Definitions (14)

  • Definition 2.1
  • Lemma 4.1
  • proof
  • Definition 5.1
  • Lemma 6.1
  • proof
  • Theorem 6.1
  • proof
  • Theorem 7.1
  • Lemma 7.1
  • ...and 4 more