[Paper Review] Global Phase Diagram of the Extended Kitaev-Heisenberg Model on Honeycomb Lattice

Jie Lou, Long Liang+2 more|arXiv (Cornell University)|Jan 28, 2015

Advanced Condensed Matter Physics3 citations

TL;DR

This study presents a global phase diagram of the extended Kitaev-Heisenberg (EKH) model on a honeycomb lattice using the multiscale entanglement renormalization ansatz (MERA), revealing eight distinct quantum phases. A rigorous spin operator dual mapping reveals hidden symmetries, showing that complex magnetic orders map to simple antiferromagnetic or ferromagnetic states, and identifies a valence bond solid (VBS) ground state in the quadro-critical region where multiple phases compete most intensely.

ABSTRACT

We study the extended Kitaev-Heisenberg (EKH) quantum spin model by adding bond-dependent off-diagonal Heisenberg term into the original KH model, which was recently proposed to describe the honeycomb Iridates. A rigorous mathematical mapping of spin operators reveals the intrinsic symmetry of the model Hamiltonian. By employing an unbiased numerical entanglement renormalization method based on tensor network ansatz, we obtain the global phase diagram containing eight distinct quantum phases. By using the dual mapping of spin operators, each of the individual magnetic phase in the global phase diagram can be clearly understood. At last, we show that a valence solid state emerges as the ground state in the quadro-critical region where multiple magnetic phases compete most intensively.

Motivation & Objective

• To map the full quantum phase diagram of the extended Kitaev-Heisenberg (EKH) model on a honeycomb lattice, including the effects of bond-dependent off-diagonal Heisenberg interactions (Γ-term).
• To understand the intrinsic symmetry of the EKH Hamiltonian through a rigorous mathematical mapping of spin operators.
• To clarify the nature of complex magnetic orders by transforming them into simpler, well-understood states (e.g., AFM/FM) via dual spin rotations.
• To investigate the emergence of quantum phases, particularly the valence bond solid (VBS), in regions of intense quantum competition.

Proposed method

• Employed the multiscale entanglement renormalization ansatz (MERA) for unbiased, numerically accurate ground state calculations on finite and infinite systems.
• Applied a dual spin rotation mapping to transform the Γ-term into equivalent Kitaev or Heisenberg interactions, revealing hidden symmetries in the Hamiltonian.
• Parameterized the model using R, θ, and φ to control relative strengths of J (Heisenberg), K (Kitaev), and Γ (off-diagonal), with J² + K² + Γ² = R².
• Used spin correlation functions and intra-/inter-plaquette correlation ratios (SC₁/SC₂) to distinguish magnetic long-range order from VBS order.
• Performed calculations on 24- and 36-site lattices to assess finite-size effects and identify phase boundaries.
• Utilized a tensor network-based ansatz to access critical regions where conventional order parameters vanish but correlations reveal VBS character.

Experimental results

Research questions

• RQ1 How does the inclusion of the bond-dependent off-diagonal Heisenberg term (Γ) alter the phase diagram of the Kitaev-Heisenberg model on a honeycomb lattice?
• RQ2 What is the role of spin operator duality in simplifying and classifying complex magnetic orders in the EKH model?
• RQ3 Where does a valence bond solid (VBS) state emerge, and what distinguishes it from conventional magnetic long-range order?
• RQ4 How do spin correlations evolve across phase transitions, particularly near the quadro-critical region?
• RQ5 To what extent do finite-size effects influence the identification of the VBS phase in the critical region?

Key findings

• The EKH model exhibits eight distinct quantum phases, including conventional magnetic orders (AFM, FM, ZZ, ST) and a valence bond solid (VBS) phase in the quadro-critical region.
• The VBS phase emerges as the ground state in regions where multiple magnetic phases compete most intensely, particularly near the points J = -K = -Γ = 1 and -J = K = Γ = 1.
• Spin correlation functions show a sharp drop in rotated spin correlations (SC_M) at the transition from FM to VBS, indicating a distinct change in entanglement and order structure.
• The ratio of intra-plaquette to inter-plaquette spin correlations (SC₁/SC₂) increases significantly in the VBS region, signaling strong local singlet formation.
• Finite-size calculations show that local spin expectation values vanish in the critical region, similar to the Kitaev spin liquid, but long-range spin correlations decay rapidly, distinguishing VBS from spin liquid behavior.
• The dual spin mapping successfully transforms complex magnetic orders into simple AFM or FM states, confirming the underlying symmetry and enabling clearer interpretation of the phase diagram.