Entropy plateaus in spin-S Kitaev Models

R R P Singh + Owen Bradley (UC Davis) J. Oitmaa (UNSW, Australia) A. Koga ( Tokyo Inst. Of Tech, Japan) D. Sen (Bangalore) Entropy plateaus in spin-S Kitaev Models

Residual Entropy in frustrated spin systems • Thermodynamics of spin-1/2 Kitaev Model • Spin-liquid in spin-S Kitaev Models • Thermodynamic Behavior and Entropy Plateaus • Ground state selection: Integer and Half-Integer S • Summary/Conclusions Outline

Frustration in Ising models leads to large ground state degeneracy Spin-ice Triangular Kagome Pauling Residual Entropy Treat constraint in each tetrahedron independently , Ramirez et al Dy2Ti2O7 Nature 1999

Quantum Spin Models A plateau must result when perturbations are small Kagome Heisenberg Elstner and Young PRB 1994 Taking Hamiltonian from:

Kitaev model (Ising (flux) + fermion variables) Thermodynamics from (classical) Monte Carlo simulation Nasu, Udagawa, Motome Plateau-like feature survives 10 % Heisenberg Coupling Phys. Rev. B 96, 144414 (2017) RRPS + J. Oitmaa

Phase Diagram Nasu, Udagawa, Motome High-temperature phase: local Intermediate phase: degenerate fermions, disordered flux: Entropy plateau Low temperature phase: flux `aligned’ Sharp phase transition on hyper-honeycomb lattices

Experiments? A fit to two phenomenological fermi distributions

Spin-S Kitaev Models Remains a spin-liquid for all S including classical limit Only nearest neighbor spin correlations are non-zero Infinite number of Conserved fluxes Huge ground state degeneracy in the classical limit

Degenerate subspace for Classical spins Ground states include Cartesian states: Baskaran, Sen, Shankar All spins aligned with one of their X,Y, or Z neighbor Any Dimer covering of the lattice gives states Dimer coverings are themselves exponential: At least an entropy of Leading order fluctuations O(S) couple spins on non-dimer SAWs Lots of zero energy modes for each SAW Lowest zero point energy from shortest SAWs --- VB pattern

Thermodynamics of spin-S Kitaev Models Oitmaa, Koga, RRPS PRB 2018 Studied by High-Temperature Expansions and Thermal Pure Quantum Method on Finite Clusters Sugiura, Shimizu PRLs Infinite T properties captured by a single random state Finite T by successive application of H on a random state E(T) Allows study of larger systems

Entropy and specific heat for spin-S Kitaev models HTE and TPQ Plateau-like features With increasing entropy Double-peaked High temperature expansions show a plateau and do not converge at lower-T

Just-like spin-half • Double-peaked C • Entropy plateau at half of max • Energy/correlation saturates below upper peak • Flux active below the peak What is the physics of increased entropy value at the plateau?

Anisotropic models Weak anisotropy: 3 peaks in heat capacity, second plateau at ln(2)/2 S=1 S=3/2 S=2

Large anisotropy case is easy to understand • Spins must align with z-neighbor: low-energy states • Gap to other states : JS • Increasing spins needs high order perturbation to resolve degeneracy: very low temperature • Separation of energy scales

What is the physics of large residual entropy in isotropic models? Classical isotropic model has continuous degeneracy Unbounded entropy --- what happens at finite S? Number of zero modes of a chain scale as D^{1/2} Semi-classical considerations explain the residual entropy

From paramagnetic to classical spin liquid behavior including entropy plateaus is smooth in S What about ground state selection and excitations? Is QSL behavior (Entanglement) also smooth in S? There may be a fundamental difference between integer and half integer spins?

Expansion around the anisotropic limit Half integer spins map on to Toric Code Integer spins map on to a single site transverse-field model Half integer and integer spin systems are very different

Baskaran, Sen, Shastry: Generalized Kitaev model One can explicitly construct Majorana Operators for half-integer spin But, they become commuting variables for integer spins Soluble Model Half integer spins map on to spin-half Kitaev Model Integer spins map on to a highly frustrated classical model as commute No entanglement! Half integer and integer spin systems are very different!

Classical spin-liquid may be smooth function of S But, quantum selection may have a non-trivial S dependence

Kitaev materials with higher spin? Ni or Cr compounds?

Summary and Conclusions Increasing spin in the Kitaev model still leaves one in a spin-liquid phase There is a low energy subspace that grows with spin leading to large entropy at intermediate T. This varies smoothly with S and corresponds to a classical spin-liquid Nature of ground state selection and excitations may be very different in integer and half-integer spin systems. Deserves further investigation. This could provide another class of candidate spin-liquid materials

Entropy plateaus in spin-S Kitaev Models