Edge Modes of Zn Symmetric Chains

Iman Mahyaeh

Published Jan 1, 2017• 0 citations

Abstract

In this Licentiate Thesis we will study the topological phases of Z2 and Z3 symmetric chains.We will present the Kitaev chain, a Z2 symmetric model of spinless fermions, and obtain allthe eigenstat ...

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Edge Modes of Zn Symmetric Chains

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Published Jan 1, 2017• 0 citations

OpenAlex

Abstract

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Study of the phase diagram of Zn symmetric chains

Iman Mahyaeh

In this thesis we study the phase diagrams of Zn symmetric chains. We start with investigating the topological phases of the Kitaev chain, a Z2 symmetric model, with long range couplings and a phase gradient. Then we go beyond the free fermion classification of topological phases and consider the effect of interactions by studying the Kitaev-Hubbard chain, incorporating a density-density interaction. Next we move on to the Z3 symmetric models and present a frustration free model with an exact three-fold degenerate ground state. In the end we present the phase diagram of a hopping model of Z3 Fock parafermions, the generalization of polarized Dirac fermions which could host at most two particles per site. The model has a pairwise hopping which is forbidden for fermions. In our studies we use analytical methods like the Lieb-Schultz-Mattis method, bosonization and conformal field theory, as well as numerical ones like exact diagonalization and the density matrix renormalization group.In this thesis we study the phase diagrams of Zn symmetric chains. We start with investigating the topological phases of the Kitaev chain, a Z2 symmetric model, with long range couplings and a phase gradient. Then we go beyond the free fermion classification of topological phases and consider the effect of interactions by studying the Kitaev-Hubbard chain, incorporating a density-density interaction. Next we move on to the Z3 symmetric models and present a frustration free model with an exact three-fold degenerate ground state. In the end we present the phase diagram of a hopping model of Z3 Fock parafermions, the generalization of polarized Dirac fermions which could host at most two particles per site. The model has a pairwise hopping which is forbidden for fermions. In our studies we use analytical methods like the Lieb-Schultz-Mattis method, bosonization and conformal field theory, as well as numerical ones like exact diagonalization and the density matrix renormalization group.

Jan 1, 20201 citation

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Edge Z3 parafermions in fermionic lattices

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Parafermion modes are non-Abelian anyons which were introduced as ${\mathbb{Z}}_{N}$ generalizations of ${\mathbb{Z}}_{2}$ Majorana states. In particular, ${\mathbb{Z}}_{3}$ parafermions can be used to produce Fibonacci anyons, laying a path towards universal topological quantum computation. Due to their fractional nature, much of the theoretical work on ${\mathbb{Z}}_{3}$ parafermions has relied on bosonization methods or parafermionic quasiparticles. In this paper, we introduce a representation of ${\mathbb{Z}}_{3}$ parafermions in terms of purely fermionic models. We establish the equivalency of a family of lattice fermionic models written in the basis of the $t\ensuremath{-}J$ model with a Kitaev-like chain supporting free ${\mathbb{Z}}_{3}$ parafermionic modes at its ends. By using density matrix renormalization group calculations, we are able to characterize the topological phase transition and study the effect of local operators (doping and magnetic fields) on the spatial localization of the parafermionic modes and their stability. Moreover, we discuss the necessary ingredients towards realizing ${\mathbb{Z}}_{3}$ parafermions in strongly interacting electronic systems.Parafermion modes are non-Abelian anyons which were introduced as ${\mathbb{Z}}_{N}$ generalizations of ${\mathbb{Z}}_{2}$ Majorana states. In particular, ${\mathbb{Z}}_{3}$ parafermions can be used to produce Fibonacci anyons, laying a path towards universal topological quantum computation. Due to their fractional nature, much of the theoretical work on ${\mathbb{Z}}_{3}$ parafermions has relied on bosonization methods or parafermionic quasiparticles. In this paper, we introduce a representation of ${\mathbb{Z}}_{3}$ parafermions in terms of purely fermionic models. We establish the equivalency of a family of lattice fermionic models written in the basis of the $t\ensuremath{-}J$ model with a Kitaev-like chain supporting free ${\mathbb{Z}}_{3}$ parafermionic modes at its ends. By using density matrix renormalization group calculations, we are able to characterize the topological phase transition and study the effect of local operators (doping and magnetic fields) on the spatial localization of the parafermionic modes and their stability. Moreover, we discuss the necessary ingredients towards realizing ${\mathbb{Z}}_{3}$ parafermions in strongly interacting electronic systems.

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The Kitaev chain model exhibits topological order that manifests as topological degeneracy, Majorana edge modes and $Z_{2}$ topological invariance of the abulk spectrum. This model can be obtained from a transverse field Ising model(TFIM) using the Jordan-Wigner transformation. TFIM has neither topological degeneracy nor any edge modes. Topological degeneracy associated with topological order is central to topological quantum computation. In this paper we will explore topological protection of the ground state manifold in the case of Majorana fermion models which exhibit $Z_{2}$ topological order. We will show that there are at least two different ways to understand this topological protection of Majorana fermion qubits: one way is based on fermionic mode operators and the other is based on anti-commuting symmetry operators. We will also show how these two different ways are related to each other. We provide a very general approach of understanding the topological protection of Majorana fermion qubits in the case of lattice Hamiltonians.The Kitaev chain model exhibits topological order that manifests as topological degeneracy, Majorana edge modes and $Z_{2}$ topological invariance of the abulk spectrum. This model can be obtained from a transverse field Ising model(TFIM) using the Jordan-Wigner transformation. TFIM has neither topological degeneracy nor any edge modes. Topological degeneracy associated with topological order is central to topological quantum computation. In this paper we will explore topological protection of the ground state manifold in the case of Majorana fermion models which exhibit $Z_{2}$ topological order. We will show that there are at least two different ways to understand this topological protection of Majorana fermion qubits: one way is based on fermionic mode operators and the other is based on anti-commuting symmetry operators. We will also show how these two different ways are related to each other. We provide a very general approach of understanding the topological protection of Majorana fermion qubits in the case of lattice Hamiltonians.

Apr 2, 20172 citations

{{\mathbb {Z}}}_2

-index of Symmetry Protected Topological Phases with Reflection Symmetry for Quantum Spin Chains

Parafermion modes are non-Abelian anyons which were introduced as

{\mathbb{Z}}_{N}

generalizations of

{\mathbb{Z}}_{2}

Majorana states. In particular,

{\mathbb{Z}}_{3}

parafermions can be used to produce Fibonacci anyons, laying a path towards universal topological quantum computation. Due to their fractional nature, much of the theoretical work on

{\mathbb{Z}}_{3}

parafermions has relied on bosonization methods or parafermionic quasiparticles. In this paper, we introduce a representation of

{\mathbb{Z}}_{3}

parafermions in terms of purely fermionic models. We establish the equivalency of a family of lattice fermionic models written in the basis of the $t\ensuremath{-}J$ model with a Kitaev-like chain supporting free

{\mathbb{Z}}_{3}

parafermionic modes at its ends. By using density matrix renormalization group calculations, we are able to characterize the topological phase transition and study the effect of local operators (doping and magnetic fields) on the spatial localization of the parafermionic modes and their stability. Moreover, we discuss the necessary ingredients towards realizing

{\mathbb{Z}}_{3}

parafermions in strongly interacting electronic systems.

Majorana zero modes, spontaneous

Z_2^{f}

symmetry breaking, and long range edge correlation in Kitaev chains: analytic solutions and density-matrix-renormalization-group study

We study Kitaev model in one-dimension with open boundary condition by using exact analytic methods for non-interacting system in the half filled limit as well as in the symmetric case of

\Delta=t

, and by using density-matrix-renormalization-group method for interacting system with nearest neighbor repulsion. We show explicitly that the emergence of Majorana zero mode is associated with the spontaneous symmetry broken of fermion number parity operator

Z_2^f

, which is

+1

for even number of fermions and

-1

for odd number of fermions. Our results are in support of recent proposal by Klassen and Wen\cite{Klassen}, who proposed such spontaneous symmetry broken in systems with Majorana zero modes. We suggest and examine an edge correlation function of Majorana fermions to characterize the long range order in the topological superconducting states and study the phase diagram of the interating Kitaev chain.

The Kitaev chain model exhibits topological order that manifests as topological degeneracy, Majorana edge modes and

Z_{2}

topological invariance of the abulk spectrum. This model can be obtained from a transverse field Ising model(TFIM) using the Jordan-Wigner transformation. TFIM has neither topological degeneracy nor any edge modes. Topological degeneracy associated with topological order is central to topological quantum computation. In this paper we will explore topological protection of the ground state manifold in the case of Majorana fermion models which exhibit

Z_{2}

topological order. We will show that there are at least two different ways to understand this topological protection of Majorana fermion qubits: one way is based on fermionic mode operators and the other is based on anti-commuting symmetry operators. We will also show how these two different ways are related to each other. We provide a very general approach of understanding the topological protection of Majorana fermion qubits in the case of lattice Hamiltonians.