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arXiv:1707.03983 · Physical review. B./Physical review. B

Exact solutions and topological phase diagram in interacting dimerized Kitaev topological superconductors

Motohiko Ezawa

Published Sep 8, 2017• 36 citations
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Abstract

It was recently shown that an interacting Kitaev topological superconductor model is exactly solvable based on two-step Jordan-Wigner transformations together with one spin rotation. We generalize this model by including the dimerization, which is shown also to be exactly solvable. We analytically determine the topological phase diagram containing seven distinct phases. It is argued that the system is topological when a fermionic many-body Majorana zero-energy edge state emerges. It is intriguing that there are two tetracritical points, at each of which four distinct phases touch.

DOI
10.1103/physrevb.96.121105

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It was recently shown that an interacting Kitaev topological superconductor model is exactly solvable based on two-step Jordan-Wigner transformations together with one spin rotation. We generalize this model by including the dimerization, which is shown also to be exactly solvable. We analytically determine the topological phase diagram containing seven distinct phases. It is argued that the system is topological when a fermionic many-body Majorana zero-energy edge state emerges. It is intriguing that there are two tetracritical points, at each of which four distinct phases touch.
It was recently shown that an interacting Kitaev topological superconductor model is exactly solvable based on two step Jordan-Wigner transformations together with one spin rotation. We generalize this model by including the dimerization, which is also exactly solvable. In this extended model there are two topological indices associated with the sublattice symmetry and the particle-hole symmetry. We analytically determine the topological phase diagram containing seven distinct topological phases. There are two tetra-critical points, at which four distinct phases touch. It is intriguing that a topological charge-density-wave state and a topological Schr\odinger-cat state emerge for strong interactions. We confirm various topological phases by examining the presence of zero-energy edge states.
We study analytically topological properties of a noninteracting modified dimerized Kitaev chain and an exactly solvable interacting dimerized Kitaev chain under open boundary conditions by analyzing two introduced edge correlation functions. The interacting dimerized Kitaev chain at the symmetry point $\mathrm{\ensuremath{\Delta}}=t$ and the chemical potential $\ensuremath{\mu}=0$ can be exactly solved by applying two Jordan-Wigner transformations and a spin rotation, which permits us to calculate the edge correlation functions analytically. We demonstrate that the two edge correlation functions can be used to characterize the trivial, Su-Schrieffer-Heeger-like topological and topological superconductor phases of both the noninteracting and interacting systems and give their phase diagrams.
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.
The Kitaev chain model with a nearest neighbor interaction U is solved exactly at the symmetry point Δ=t and chemical potential μ=0 in an open boundary condition. By applying two Jordan-Wigner transformations and a spin rotation, such a symmetric interacting model is mapped onto a noninteracting fermion model, which can be diagonalized exactly. The solutions include a topologically nontrivial phase at |U|<t and a topologically trivial phase at |U|>t. The two phases are related by dualities. Quantum phase transitions in the model are studied with the help of the exact solution.
The fermionic and bosonic zero modes of the one-dimensional (1D) interacting Kitaev chain at the symmetric point are unveiled. The many-body structures of the Majorana zero modes in the topological region are given explicitly by carrying out a perturbation expansion up to infinite order. We also give the analytic expressions of the bosonic zero modes in the topologically trivial phase. Our results are generalized to the hybrid fermion system comprised of the interacting Kitaev model and the Su–Schrieffer–Heeger (SSH) model, in which we show that these two types of zero modes can coexist in a certain region of its phase diagram.
A family of two-dimensional (2D) spin-1/2 models have been constructed to realize Kitaev’s sixteen-fold way of anyon theories. Defining a one-dimensional (1D) path through all the lattice sites, and performing the Jordan-Wigner transformation with the help of the 1D path, we find that such a spin-1/2 model is equivalent to a model with \nu ν species of Majorana fermions coupled to a static \mathbb{Z}_2 ℤ 2 gauge field. Here each species of Majorana fermions gives rise to an energy band that carries a Chern number \mathcal{C}=1 𝒞 = 1 , yielding a total Chern number \mathcal{C}=\nu 𝒞 = ν . It has been shown that the ground states are three (four)-fold topologically degenerate on a torus, when \nu ν is an odd (even) number. These exactly solvable models can be achieved by quantum simulations.