[Paper Review] A Spin-Orbital Singlet and Quantum Critical Point on the Diamond Lattice: FeSc2S4

Gang Chen, Leon Balents+1 more|arXiv (Cornell University)|Oct 3, 2008

Topological Materials and Phenomena6 citations

TL;DR

This paper proposes that FeSc2S4 is near a quantum critical point (QCP) between a spin-orbital singlet (SOS) phase driven by spin-orbit coupling and a magnetically and orbitally ordered phase driven by exchange interactions. Using a symmetry-constrained spin-orbital Hamiltonian with nearest- and next-nearest-neighbor interactions, the authors show that the system exhibits a broad quantum critical regime, explaining the observed spin-orbital liquid behavior down to 50 mK and the small energy gap (~2 K) in neutron and NMR data.

ABSTRACT

We present a theory of spin and orbital physics in the A-site spinel compound FeSc2S4, which experimentally exhibits a broad "spin-orbital liquid" regime. A spin-orbital Hamiltonian is derived from a combination of microscopic consideration and symmetry analysis. We demonstrate a keen competition between spin-orbit interactions, which favor formation of a local "Spin-Orbital Singlet" (SOS), and exchange, which favors magnetic and orbital ordering. Separating the SOS from the ordered state is a quantum critical point (QCP). We argue that FeSc2S4 is close to this QCP on the SOS side. The full phase diagram of the model includes a commensurate-incommensurate transition within the ordered phase. A variety of comparison to and suggestion for experiments are discussed.

Motivation & Objective

• To explain the broad spin-orbital liquid regime in FeSc2S4, which persists down to 50 mK with no long-range order.
• To identify the microscopic origin of the quantum critical point (QCP) separating a spin-orbital singlet (SOS) phase from a magnetically and orbitally ordered phase.
• To derive a symmetry-allowed spin-orbital Hamiltonian for the diamond lattice of FeSc2S4, incorporating both nearest- and next-nearest-neighbor exchange interactions.
• To connect theoretical predictions to experimental observations, including neutron scattering, NMR relaxation, and specific heat data.
• To suggest experimental routes to drive the system across the QCP, such as pressure or magnetic field tuning.

Proposed method

• Derive a Kugel-Khomskii-type spin-orbital Hamiltonian using symmetry analysis and microscopic considerations, focusing on superexchange via eg orbitals.
• Construct a spin-orbital Hamiltonian with exchange couplings J1, J2 (nearest- and next-nearest-neighbor), and spin-orbit coupling λ, including orbital pseudospin terms.
• Use perturbation theory to map the effective spin-orbital Hamiltonian from the Hubbard model, integrating out intermediate states on X and B sites.
• Analyze the phase diagram via classical energy minimization, identifying commensurate and incommensurate spiral spin and orbital orderings.
• Compute the low-energy excitation spectrum in the SOS phase using a small-x expansion in the exchange coupling, yielding a gap formula.
• Compare theoretical predictions with neutron scattering, NMR relaxation (1/T1), and specific heat data to validate the QCP scenario.

Experimental results

Research questions

• RQ1 What is the microscopic origin of the broad spin-orbital liquid regime in FeSc2S4, and why does it persist to such low temperatures?
• RQ2 How does the competition between spin-orbit coupling (favoring SOS) and exchange interactions (favoring order) lead to a quantum critical point?
• RQ3 What is the nature of the phase transition between the spin-orbital singlet and the ordered phases, and what is the critical point's location in parameter space?
• RQ4 Why is the observed energy gap (~2 K) so small compared to the Curie-Weiss temperature (~45 K), and how does this relate to quantum criticality?
• RQ5 Can the observed power-law specific heat (T^2.5) and non-monotonic 1/T1 behavior be explained by proximity to a QCP?

Key findings

• The system is near a quantum critical point (QCP) separating a spin-orbital singlet (SOS) phase from a magnetically and orbitally ordered phase, with the SOS phase stabilized by spin-orbit coupling.
• The low-energy excitation spectrum in the SOS phase is gapped, with a minimum energy gap Δ ≈ 2 K, consistent with neutron scattering and NMR data.
• The energy gap is given by Δ = λ - 8(J2 + K2) - (J1 + K1)² / [2(J2 + K2)], with the gap closing at the QCP when x → 0.
• The specific heat exhibits a T^2.5 power-law dependence between 0.2 K and 2 K, suggesting a crossover from two-level systems to quantum critical magnetic contributions.
• The uniform magnetic susceptibility remains large at low temperatures, indicating strong spin-orbit coupling effects and proximity to the QCP.
• Theoretical analysis predicts a commensurate-incommensurate transition within the ordered phase, with incommensurate order favored for x > xc and commensurate order for xc < x < xc1.