[Paper Review] A $Γ$-matrix generalization of the Kitaev model

Congjun Wu, Daniel P. Arovas+1 more|arXiv (Cornell University)|Nov 10, 2008

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper generalizes the Kitaev model from Pauli matrices to higher-rank Clifford algebras using $\Gamma$-matrices, constructing a solvable spin model on a decorated square lattice with coordination number five. The ground state breaks time-reversal symmetry spontaneously and realizes a topologically nontrivial chiral spin liquid with gapless chiral edge modes; on the 3D diamond lattice, it supports gapless Dirac-like excitations and gapped topological insulating states, extending the scope of exactly solvable topological phases beyond the original Kitaev model.

ABSTRACT

We extend the Kitaev model defined for the Pauli-matrices to the Clifford algebra of $Γ$-matrices, taking the $4 imes 4$ representation as an example. On a decorated square lattice, the ground state spontaneously breaks time-reversal symmetry and exhibits a topological phase transition. The topologically non-trivial phase carries gapless chiral edge modes along the sample boundary. On the 3D diamond lattice, the ground states can exhibit gapless 3D Dirac cone-like excitations and gapped topological insulating states. Generalizations to even higher rank $Γ$-matrices are also discussed.

Motivation & Objective

• To extend the Kitaev model, originally based on Pauli matrices, to higher-rank Clifford algebras using $\Gamma$-matrices.
• To construct a solvable spin model on a decorated square lattice with coordination number five, realizing a chiral spin liquid phase.
• To explore topological phases on the 3D diamond lattice, including gapless Dirac cone-like excitations and gapped topological insulators.
• To generalize the framework to even higher-rank $\Gamma$-matrices, identifying states with a 'time-reversal-like' symmetry.

Proposed method

• Represent the $\Gamma$-matrices as products of Majorana fermions, embedding them in a $2n \times 2n$ Clifford algebra with $\{\Gamma^a, \Gamma^b\} = 2\delta^{ab}$.
• Define a Hamiltonian $\mathcal{H} = -\sum_{\langle ij\rangle} J_{ij} \Gamma^a_i \Gamma^a_j$ on a lattice with $z=2n-1$ coordination, where each link carries a color label $a$.
• Map the spin interactions to a $\mathbb{Z}_2$ gauge theory via $u_{ij} = i\xi^a_i \xi^a_j$, with $u_{ij}^2 = 1$ and $[u_{ij}, u_{kl}] = 0$, enabling exact solvability.
• Use the $4\times4$ $\Gamma$-matrix representation to model a spin-$3/2$ system with anisotropic interactions involving only spin-quadrupole operators.
• Analyze the 3D diamond lattice model by tuning couplings to break time-reversal symmetry explicitly, yielding a gapless spin liquid with a 3D Dirac cone spectrum.
• Apply the Fu-Kane parity criterion to classify topological insulating states on the diamond lattice based on $\gamma^{45}$ parity eigenvalues of occupied states.

Experimental results

Research questions

• RQ1 Can the Kitaev model be generalized beyond Pauli matrices to higher-rank Clifford algebras using $\Gamma$-matrices?
• RQ2 Does a $4\times4$ $\Gamma$-matrix model on a decorated square lattice exhibit spontaneous time-reversal symmetry breaking and topological order?
• RQ3 What are the low-energy excitations and edge modes in the 2D chiral spin liquid phase of the generalized model?
• RQ4 Can a 3D counterpart of the Kitaev model on the diamond lattice host gapless Dirac-like excitations or gapped topological insulators?
• RQ5 How do time-reversal-like symmetries emerge in higher-rank $\Gamma$-matrix models, and what topological phases do they support?

Key findings

• The $4\times4$ $\Gamma$-matrix model on a decorated square lattice realizes a topologically nontrivial chiral spin liquid with spontaneously broken time-reversal symmetry, despite individual spin-quadrupole operators being TR-invariant.
• The ground state exhibits gapless chiral edge modes along the sample boundary, confirming its topological nature.
• On the 3D diamond lattice, the model supports a gapless spin liquid phase with a 3D Dirac cone-like spectrum when time-reversal symmetry is explicitly broken.
• By tuning coupling constants, the system can be driven into a gapped topological insulator phase, with the topological invariant determined by the parity eigenvalues of $\gamma^{45}$.
• For $J_4 \neq J_{1,2,3}$, a gap opens at the Dirac points, and the system becomes a topological insulator for $J_4 > J$, hosting surface modes with an odd number of Dirac cones.
• The generalization to higher-rank $\Gamma$-matrices reveals topological spin liquid states with a well-defined 'time-reversal-like' symmetry, suggesting broader classes of solvable topological phases.