Engineering Topological Mott Phases 

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The Honeycomb Iridates

The hexagonal iridates Na2IrO3 and Li2IrO3 realize a layered structure consisting of a hon- eycomb lattice of Ir4+ ions, and they provide a concrete example of the full orbital degeneracy lift with a maximally quantum effective spin-1/2 Hamiltonian. Both compounds appear to be in the strong Mott regime. As shown by Jackeli and Khaliullin [43, 44], the edge sharing octa- hedral structure and the structure of the entangled Jeff = 1/2 orbitals leads to a cancellation of the usually dominant antiferromagnetic oxygen-mediated exchange interactions. A sub- dominant term is generated by Hund’s coupling, which takes the form of a highly anisotropic Kitaev exchange coupling: HK = −K Ø α=x,y,z Ø éi,jê∈α Sα i Sα j , (1.4.5).
where Si are the effective spin-1/2 operators and α = x, y, z labels both spin components and the three orientations of links on the honeycomb lattice. This particular Hamiltonian realizes the exactly solvable model of the quantum spin liquid phase proposed by Alexei Kitaev[15], which describes the fractionalization of the spins into Majorana fermions, stemming from the geometry and entanglement in the strong spin-orbit coupling limit. In Chapter 3 section 3.2, we have shown how this geometry and orbital entanglement brings extra symmetry, endowing the Kitaev model a self-duality point with extra symmetry. In antiferromagnets, spin-orbit coupling will remove accidental degeneracy and favor order via the Dzyaloshinskii-Moriya interaction, while the Kitaev model is a counterexample, in which spin-orbit coupling can suppress ordering. The experimental studies through neutron scattering and other studies show that the ground state of Na2IrO3 displays a collinear magnetic order, the zigzag state with a four-sublattice structure, arising from possible Heisenberg coupling [116, 117]. However, the Kitaev coupling might be much larger in Li2IrO3 and the system may be closer to the quantum spin liquid phase. There has also been proposition in ultra cold atoms for the the realization of the Kitaev model [46].

Topological Insulator Phase

In this section, we explore the physics in the weakly correlated regime, namely physics in the limit of U ≪ t, t′. The normal hopping between nearest-neighbours (NN) with strength t gives a graphene band structure with Dirac cones at the corners of the first-Brillouin zone, while the spin-orbit coupling with strength t′ opens a gap for the band electrons at the Dirac cones. The spin-orbit coupling introduces into the system opposite effective magnetic fields for spins with opposite polarizations pointing along different directions on different links. In the presence of time-reversal symmetry (TRS), Kramers Theorem states that system of spin 1/2 with time-reversal symmetry (TRS) is necessarily doubly degenerate, with one sector odd under TRS, and another even under TRS. The symmetry group related to TRS is Z2, and the gaplessness of the edge mode is ensured by the TRS in that any processes opening a gap for the edge mode breaks the TRS. The topological aspect of the Kane-Mele model can be easily illustrated by studying the spin transport on the edge knowing that the Pauli matrix σz commutes with the Hamiltonian, indicating well-defined spin current along the Z direction, in other words quantum spin Hall effect.
The anisotropic spin-orbit coupling model in the weakly correlated regime describes quan- tum spin Hall effect with the same Z2 topological index as the Kane-Mele model. Neglecting the Hubbard interaction in the first place, we diagonalize the tight binding model H0 by Fourier transformation: 2.1.1: H0 = Ø <i,j> tc†iσcjσ + Ø ≪i,j≫ it′σα σσ′c†iσcjσ′ = Ø þk $†þ k h(þk)$þk.

Intermediate Interaction Region – Mott Transition

When the Hubbard interaction becomes significant enough compared to the electron hopping energy for a system with half-filling, electron charges tend to be gradually localized by the interaction because the hopping processes bring about virtual states of a doubly occupied site, which is energetically very costly. However, the second order processes in which two electrons with different spins exchange their positions is allowed, and when the Hubbard interaction approaches infinity compared to electron hopping, charges are completely localized while the only reminiscent processes are the super-exchange processes of the spins, which gives rise to the magnetic order, as if the charge and spin of the electrons were totally separated [34]. The underlying physics describing this transition from free electrons to charge localization and spin charge separation is the Mott transition. One way to describe this transition is the slave particle representation, in which electrons are represented as a charge particle (chargeon) and a spin particle (spinon) glued together by a gauge field[35]. In the weak coupling limit, the chargeon is in a gapless superfluid phase while at certain critical interaction strength Uc the chargeon acquires a gap, and the correlation length of the chargeon decays exponentially [22].
However, for a topological insulator in which the bulk is a gapped insulator and the edge is metallic, the bulk is gapped by the spin-orbit coupling in the weakly coupling limit and gapped by the interaction in the Mott phase, and the Mott transition is manifested rather on the edge by the disappearance of the helical edge modes[222]. The single-electron gap does not close at the Mott transition and the density of states now centers around the Hubbard energy in the spectral function [22], as an embodiment of the Coulomb blockade. A gauge field will however emerge in this spin-charge separation physics describing the confining force between the charge and the spin above the Mott transition. Gauge field in 2 + 1D has monopoles as quantum tunneling processes or gauge fluctuations, and the particle corresponding to such processes are instantons that can be described as a the classical plasma. By this analogy, there exists a force between the monopoles. The nature of this confining force might determine whether above the Mott critical point the system is in a spin liquid phase [95] or already in a magnetically ordered phase. On the one hand, if the monopoles are confined, then gauge field is in a dielectric phase with equal positive and negative topological charges for the monopoles, spinons excitation are deconfined and the system is probably in the spin liquid phase; on the other hand, when the monopoles are in the deconfined plasma phase, monopoles proliferate and spinons are confined, leading to a long rang magnetic order.

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Slave Rotor Representation for the Mott Transition

The U(1) slave-rotor representation [130, 131] consists of labelling the 4 state Hilbert space by angular momentum: |↑êe = |↑ês |0êθ, |↓êe = |↓ês |0êθ, |↑↓êe = |↑↓ês |1êθ and |φêe = |φês |−1êθ. The creation of a physical electron is the creation of a spin in the spinon Hilbert space accompanied by raising the angular momentum in the rotor Hilbert space, while the measure of the number of electron is the measure of the angular momentum: c†σ = f†σeiθ cσ = fσe−iθ.

Table of contents :

1 Introduction 
1.1 Topology in Condensed Matter
1.1.1 Quantum Hall System
1.1.2 Haldane Model and Chern Number
1.1.3 Kane-Mele Model and Z2 Topological Invariant
1.2 Mott Physics
1.2.1 Doped Mott Insulators
1.3 Frustrated Magnetism
1.3.1 Geometrical Frustration
1.3.2 Order by Disorder
1.3.3 Spin Liquid
1.3.4 Kane-Mele-Hubbard Model
1.4 Introduction to Iridate System
1.4.1 Balents’ Diagram
1.4.2 The Honeycomb Iridates
1.5 Doped Honeycomb Iridates
1.6 Summary
2 Iridates on Honeycomb Lattice at Half-filling 
2.1 Topological Insulator Phase
2.1.1 Numerical Diagonalization
2.1.2 Edge State Solution via Transfer Matrix
2.2 The Frustrated Magnetism in Strong Coupling limit
2.2.1 Néel Phase for J1 > J2
2.2.2 Non-Colinear Spiral Phase for J1 < J2
2.2.3 Phase Transition at J1 = J2
2.3 Intermediate Interaction Region – Mott Transition
2.3.1 Slave Rotor Representation for the Mott Transition
2.3.2 Gauge Fluctuation Upon Mott Transition
2.3.3 Spin Texture upon Insertion of Flux
2.4 Lattice Gauge Field by Construction of Loop Variables
2.5 Spin Texture under Two Adjacent Monopoles
2.6 Conclusion
3 Doping Iridates on the Honeycomb Lattice – t − J Model 
3.1 Introduction
3.2 Duality between Heisenberg and Kitaev-Heisenberg model
3.2.1 Duality at Half-filling
3.2.2 Duality beyond Half-filling
3.3 Exact Diagonalization on one Plaquette – Triplet Pairings
3.3.1 Half-filling
3.3.2 Doped System
3.4 Band Structure of the Spin-Orbit Coupling System
3.5 FFLO Superconductivity
3.5.1 The Spin-Orbit Coupling Limit t = J1 = 0
3.5.2 Near the Spin-Orbit Coupling Limit t, J1 → 0
3.6 Numerical Proofs of the FFLO Superconductivity
3.7 Conclusion
4 Engineering Topological Mott Phases 
4.1 RKKY Interaction
4.2 Haldane Mass Induced by the RKKY Interaction
4.3 Mott Transition Induced by the RKKY Interaction
4.4 Conclusion
5 Conclusion 
A Annexe 
A.1 Loop Variables Construction: Curl and Divergence on a Lattice

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