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Constructing a variational ground state of matter fermions coupled to a vison pair in Kitaev's honeycomb model

T. Grace, A. Principi

TL;DR

This work develops a concise variational framework to construct explicit ground-state approximations for Kitaev's honeycomb model with a neighbouring vison pair by keeping only the largest singular values of the transformation between vison-pair and flux-free states. The resulting ansatz yields analytic expressions for the vison-pair matter state, parameterised by $a_{\mathbf{k}}$, $b_{\mathbf{k}}$, and $\theta$, and gives a ground-state energy shift $\Delta \approx 0.272|K|$ (within $3\%$ of the exact $0.263|K|$). Physical quantities such as link-energy changes and vison-pair hopping calculated from the variational state closely match exact-diagonalisation results on finite lattices, with errors around 5–6%. The method enables studying much larger systems than feasible with ED, and can be extended to more complex flux configurations, longer vison strings, and perturbations (e.g., Heisenberg, Gamma, Zeeman terms), offering a tractable route to semi-analytic insights into vison dynamics in perturbed Kitaev models.

Abstract

We develop a new method to construct simple and explicit variational approximations for the ground state of Kitaev's honeycomb model with a non-trivial Z2 flux configuration consisting of a pair of visons on neighbouring plaquettes. The method consists of retaining only the largest singular values of the generator of the transformation between the vison-pair and flux-free ground states. We compare physical quantities calculated using the approximate state to those obtained by extrapolating results of exact diagonalisation of finite lattices, finding them to be in very good agreement. We discuss ways to extend the method to more complicated flux configurations.

Constructing a variational ground state of matter fermions coupled to a vison pair in Kitaev's honeycomb model

TL;DR

This work develops a concise variational framework to construct explicit ground-state approximations for Kitaev's honeycomb model with a neighbouring vison pair by keeping only the largest singular values of the transformation between vison-pair and flux-free states. The resulting ansatz yields analytic expressions for the vison-pair matter state, parameterised by , , and , and gives a ground-state energy shift (within of the exact ). Physical quantities such as link-energy changes and vison-pair hopping calculated from the variational state closely match exact-diagonalisation results on finite lattices, with errors around 5–6%. The method enables studying much larger systems than feasible with ED, and can be extended to more complex flux configurations, longer vison strings, and perturbations (e.g., Heisenberg, Gamma, Zeeman terms), offering a tractable route to semi-analytic insights into vison dynamics in perturbed Kitaev models.

Abstract

We develop a new method to construct simple and explicit variational approximations for the ground state of Kitaev's honeycomb model with a non-trivial Z2 flux configuration consisting of a pair of visons on neighbouring plaquettes. The method consists of retaining only the largest singular values of the generator of the transformation between the vison-pair and flux-free ground states. We compare physical quantities calculated using the approximate state to those obtained by extrapolating results of exact diagonalisation of finite lattices, finding them to be in very good agreement. We discuss ways to extend the method to more complicated flux configurations.

Paper Structure

This paper contains 13 sections, 51 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of Kitaev's honeycomb model
  3. Matter fermion ground state in the presence of a vison pair
  4. Variational approximation of the vison pair state
  5. Comparison to exact results
  6. Link energy around vison pairs
  7. Hopping of vison pairs
  8. Discussion
  9. Minimisation of the ground state energy
  10. Calculation of hopping parameters
  11. Heisenberg Interaction
  12. Gamma Interactions
  13. Zeeman Interactions

Figures (2)

  • Figure 1: The geometry of Kitaev's honeycomb model. $\mathbf{a}_1 = \frac{1}{2}(1,\sqrt{3})$ and $\mathbf{a}_2=\frac{1}{2}(-1,\sqrt{3})$ are the primitive lattice vectors, while $\bm r_\alpha$ ($\alpha=x,y,z$) denotes the vectors connecting a site of sublattice A to its three B-type nearest neighbours. Dimensionless units are used such that $|\mathbf{a}_1|=|\mathbf{a}_2|=1$.
  • Figure 2: Panel (a): The change in link energy around the vison pair calculated using the variational Ansatz in Eq. (\ref{['eq:pair_variational_ansatz']}). Panel (b): The difference between the link energies calculated using the variational Ansatz (\ref{['eq:pair_variational_ansatz']}) and exact diagonalisation on a $58\times58$ lattice with periodic boundaries. Note the change in scale compared to Fig. (\ref{['fig:link_energy']}) by about one order of magnitude. In both panels, visons are located on the shaded plaquettes marked with "$\pi$".