Kitaev Models R R P Singh Owen Bradley UC Davis J Oitmaa UNSW Australia A Koga Tokyo Inst Of Tech Japan D Sen Bangalore ID: 814414
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Slide1
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)
Outline
Residual Entropy in frustrated spin systemsThermodynamics of spin-1/2 Kitaev ModelSpin-liquid in spin-S Kitaev
ModelsThermodynamic Behavior and Entropy Plateaus
Ground state selection: Integer and Half-Integer SSummary/Conclusions
Slide3Frustration in Ising
models leads to large ground state degeneracy
Triangular
Kagome
Spin-ice
Pauling Residual Entropy
Treat
constraint in each tetrahedron independently
,
Ramirez et al Dy2Ti2O7 Nature 1999
Slide4Quantum Spin Models
Kagome HeisenbergElstner
and Young PRB 1994
A plateau must result when perturbations are small
Taking Hamiltonian from:
Slide5Kitaev model (Ising (flux) + fermion variables)
Thermodynamics from (classical) Monte Carlo simulation
Nasu,
Udagawa, Motome
RRPS + J.
Oitmaa
Phys. Rev. B 96, 144414 (2017)
Plateau-like feature survives 10 % Heisenberg Coupling
Slide6Phase Diagram
High-temperature phase: localIntermediate phase: degenerate fermions, disordered flux: Entropy plateauLow temperature phase: flux `aligned’
Sharp phase transition on hyper-honeycomb lattices
Nasu
, Udagawa,
Motome
Slide7Experiments?
A fit to two phenomenological fermi distributions
Spin-S Kitaev Models
Infinite number of Conserved fluxes
Huge ground state degeneracy in the classical limit
Remains a spin-liquid for all S including classical limit
Only nearest neighbor spin correlations are non-zero
Slide9Degenerate subspace for Classical spins Ground states include Cartesian states
: Baskaran, Sen, ShankarAll spins aligned with one of their X,Y, or Z neighborAny 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
Slide10Thermodynamics 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
E(T)
Infinite T properties captured by a single random state
Finite T by successive application of H on a random stat
e
Allows study of larger systems
Slide11Entropy and specific heat for spin-S
Kitaev
models
HTE and TPQDouble-peakedPlateau-like features
With increasing entropyHigh temperature expansions show a plateau and do not converge at lower-T
Slide12Just-like spin-half
Double-peaked C
Entropy plateau at half of max
Energy/correlation saturates below upper peakFlux active below the peak
What is the physics of increased entropy value at the plateau?
Slide13Anisotropic models
S=1
S=3/2
S=2
Weak anisotropy: 3 peaks in heat capacity, second plateau at ln(2)/2
Slide14Large anisotropy case is easy to understand
Spins must align with z-neighbor: low-energy statesGap to other states : JS
Increasing spins needs high order perturbation to resolve degeneracy: very low temperatureSeparation 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
Slide16From paramagnetic to classical spin liquid behavior including entropy plateaus is smooth in SWhat 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?
Slide17Expansion 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
Slide18Baskaran, Sen, Shastry: G
eneralized Kitaev 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!
One can explicitly construct
Majorana
Operators for half-integer spin
But, they become commuting variables for integer spins
Soluble Model
Slide19Classical spin-liquid may be smooth function of SBut, quantum selection may have a non-trivial S dependence
Slide20Kitaev materials with higher spin?
Ni or Cr compounds?
Slide21Summary and Conclusions
Increasing spin in the Kitaev model still leaves one in a spin-liquid phaseThere 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-liquidNature 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