Entropy plateaus in spin-S - PowerPoint Presentation

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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

Frustration 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

Quantum Spin Models

Kagome HeisenbergElstner

and Young PRB 1994

A plateau must result when perturbations are small

Taking Hamiltonian from:

Kitaev 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

Phase 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

Experiments?

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

Degenerate 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

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

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

Entropy 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

Just-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?

Anisotropic models

S=1

S=3/2

S=2

Weak anisotropy: 3 peaks in heat capacity, second plateau at ln(2)/2

Large 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

From 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?

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: 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

Classical spin-liquid may be smooth function of SBut, 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 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