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arXiv:0811.1380 · Physical Review B

Γ -matrix generalization of the Kitaev model

Congjun Wu, Daniel P. Arovas, Hsiang-Hsuan Hung

Published Apr 22, 2009• 77 citations
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Abstract

We extend the Kitaev model defined for the Pauli matrices to the Clifford algebra of $\ensuremath{\Gamma}$ matrices, taking the $4\ifmmode\times\else\texttimes\fi{}4$ 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 nontrivial phase carries gapless chiral edge modes along the sample boundary. On the three-dimensional (3D) diamond lattice, the ground states can exhibit gapless 3D Dirac-cone-like excitations and gapped topological insulating states. Generalizations to even higher rank $\ensuremath{\Gamma}$ matrices are also discussed.

DOI
10.1103/physrevb.79.134427

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We extend the Kitaev model defined for the Pauli matrices to the Clifford algebra of \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ matrices, taking the 4\ifmmode×\else\texttimes\fi44\ifmmode\times\else\texttimes\fi{}44\ifmmode×\else\texttimes\fi4 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 nontrivial phase carries gapless chiral edge modes along the sample boundary. On the three-dimensional (3D) diamond lattice, the ground states can exhibit gapless 3D Dirac-cone-like excitations and gapped topological insulating states. Generalizations to even higher rank \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ matrices are also discussed.
We have proposed an exactly solvable model defined on 2D decorated lattices of two types. The ground-state phase diagram of the system includes different topological phases with gapless chiral edge states. We show that two types of chiral spin liquid with gapless edge modes are realized on lattices with different symmetry. The phase transition between the topological phase with chiral gapped (Chern number zero) and the topological phase with chiral gapless edge modes (Chern number ) occurs in the model on the square (symmetric) decorated lattice. On the rectangular (asymmetric) decorated lattice the topological phase is defined by a chiral gapless (gapped) edge mode in the x ( y ) direction and a chiral gapped (gapless) edge mode in another y ( x ) direction. We show that a Kitaev model on a decorated asymmetric square lattice exhibits the quantum phase transition between topological phases with equal Chern numbers.
A time-reversal invariant Kitaev-type model is introduced in which spins (Dirac matrices) on the square lattice interact via anisotropic nearest-neighbor and next-nearest-neighbor exchange interactions. The model is exactly solved by mapping it onto a tight-binding model of free Majorana fermions coupled with static Z2{\mathbb{Z}}_{2}Z2​ gauge fields. The Majorana fermion model can be viewed as a model of time-reversal-invariant superconductor and is classified as a member of symmetry class DIII in the Altland-Zirnbauer classification. The ground-state phase diagram has two topologically distinct gapped phases which are distinguished by a Z2{\mathbb{Z}}_{2}Z2​ topological invariant. The topologically nontrivial phase supports both a Kramers' pair of gapless Majorana edge modes at the boundary and a Kramers' pair of zero-energy Majorana states bound to a 0-flux vortex in the $\ensuremath{\pi}$-flux background. Power-law decaying correlation functions of spins along the edge are obtained by taking the gapless Majorana edge modes into account. The model is also defined on the one-dimension ladder, in which case again the ground-state phase diagram has Z2{\mathbb{Z}}_{2}Z2​ trivial and nontrivial phases.
We present an exactly solvable spin-orbital model based on the Gamma-matrix generalization of a Kitaev-type Hamiltonian. In the presence of small magnetic fields, the model exhibits a critical phase with a spectrum characterized by topologically protected Fermi points. Upon increasing the magnetic field, Fermi points carrying opposite topological charges move toward each other and annihilate at a critical field, signaling a phase transition into a gapped phase with trivial topology in three dimensions. On the other hand, by subjecting the system to a staggered magnetic field, an effective time-reversal symmetry essential to the existence of three-dimensional topological insulators is restored in the auxiliary free fermion problem. The nontrivial topology of the gapped ground state is characterized by an integer winding number and manifests itself through the appearance of gapless Majorana fermions confined to the two-dimensional surface of a finite system.
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Topological phases of matter describe systems beyond the paradigm of spontaneous symmetry breaking, giving rise to unconventional phenomena with possible applications in topological quantum computing. Almost 40 years after their discovery, topological phases remain among the most active research fields of condensed matter physics. As evidenced by the recent discovery of higher order topological insulators (HOTIs), novel physics can emerge even in non-interacting systems. Theoretical investigations of strongly correlated systems are extremely difficult due to the exponential growth of the quantum-mechanical Hilbert space and frequently require the application of approximative methods such as model wave functions. Tensor network states (TNS) are a class of variational wave functions which allow an efficient encoding of relevant quantum many body states and which form the basis for very successful numerical algorithms. Projected entangled pair states (PEPS), a class of TNS in two and higher dimensions, permit the exact representation of many interacting topological phases with time reversal symmetry. However, the description of experimentally relevant chiral topological phases like the integer and fractional quantum Hall effects is much more subtle using the TNS framework. The first part of this thesis focuses on the description of chiral topological phases using PEPS. Firstly, we study a previously proposed chiral PEPS that was conjectured to possess anyonic excitations. By a careful analysis of its symmetries we are able to align important entanglement observables more closely with the expected universal behavior. This highlights that efficient PEPS can possess characteristic properties of chiral topological phases. Secondly, we focus on the issue that all known chiral PEPS have algebraic bulk correlation functions and therefore cannot be the ground states of gapped local Hamiltonians. Since this problem arises already for non-interacting chiral topological phases, we focus on two examples of such systems and show that their ground states can be represented exactly by efficient PEPS in a hybrid lattice with one momentum-space direction. After an inverse Fourier transform, the PEPS with only real-space coordinates requires an exponentially growing number of parameters for an exact representation. This provides a concrete illustration of the impossibility to encode gapped chiral phases exactly with efficient PEPS. Finally, we analyze a model wave function for a three-dimensional (3D) HOTI with strong intrinsic correlations using large-scale variational Monte Carlo simulations. This wave function is obtained by projection of two copies of a non-interacting HOTI with chiral hinge states. We characterize the gapless hinge states of the interacting system and show that they are of the same nature as the edge states of the 1/21/21/2 Laughlin state. Surprisingly, the gapped surfaces host a two-dimensional (2D) phase whose topological entanglement entropy is half of that of the 1/21/21/2 Laughlin state. Such a value cannot be obtained by any of the known 2D topological orders, showing a clear departure not only from the Laughlin 1/21/21/2 physics but from conventional 2D topological order. This demonstrates that 3D topological phases can host rich phenomena that are not yet fully understood.