1. 1 Spin Freezing in Geometrically Frustrated Antiferromagnets with Weak Bond Disorder Tim Saunders Supervisor: John Chalker

2. 2 Talk Outline Low Temperature Properties of Clean Geometrically Frustrated Antiferromagnets Disorder: Experimental Background The Single Tetrahedron with Bond Disorder Nature of the Frozen State Monte Carlo Simulations: Spin-Freezing Phase Conclusions

3. 3 What is Frustration? Lattice geometry versus magnetic interactions Lattice geometry versus magnetic interactions Examples: Examples: PyrochloreKagome Insulating magnetic materials Insulating magnetic materials

4. 4 Ground State Properties Classical nearest-neighbour antiferromagnetic Hamiltonian: Ground state condition: sum of the spins on each unit is zero. Ground state condition: sum of the spins on each unit is zero. Groundstate is macroscopically degenerate Groundstate is macroscopically degenerate Main distinction between frustrated and unfrustrated magnets. Main distinction between frustrated and unfrustrated magnets. Can be rewritten as

5. 5 Ground State Degeneracy K=nN ground state constraints K=nN ground state constraints (n=no. of spin components, N = no. of tetrahedra) (n=no. of spin components, N = no. of tetrahedra) F=(qN/2)(n-1) d.o.f. for spins F=(qN/2)(n-1) d.o.f. for spins (q= no. of spins per unit). (q= no. of spins per unit). Ground state dimension: F-K = N[n(q-2)-q]/2 Ground state dimension: F-K = N[n(q-2)-q]/2 Heisenberg spins on pyrochlore lattice (q=4): Heisenberg spins on pyrochlore lattice (q=4): (F-K)/N = 1 G.S. configuration can be freely rotated into another G.S. configuration with no energy cost G.S. configuration can be freely rotated into another G.S. configuration with no energy cost Macroscopic degeneracy Macroscopic degeneracy

6. 6 Ground State Parameterisation Ground state equivalent to divergence-free condition on ‘magnetic flux’. Ground state equivalent to divergence-free condition on ‘magnetic flux’. Map spin components to flux lines on direct lattice – a diamond lattice for the pyrochlore. Map spin components to flux lines on direct lattice – a diamond lattice for the pyrochlore. The direct lattice must be bi-partite The direct lattice must be bi-partite The flux fields obey The flux fields obey Define vector potential A α - parameterises ground state. Define vector potential A α - parameterises ground state. Each ground state configuration is equivalent to a unique arrangement of flux lines. Used later to discuss effect of quenched disorder Each ground state configuration is equivalent to a unique arrangement of flux lines. Used later to discuss effect of quenched disorder

7. 7 Experimental Background: Spin Freezing Pure system paramagnetic down to very low temperature. Pure system paramagnetic down to very low temperature. |θ c-w | ~ 450K, T f ~ 4K Nature of transition similar to spin glasses. Nature of transition similar to spin glasses. Susceptibility obeys Curie-Weiss law to temperatures <<|θ c-w | Susceptibility obeys Curie-Weiss law to temperatures <<|θ c-w | Typically, at T f <<|θ c-w |, there is a transition to a frozen state. Typically, at T f <<|θ c-w |, there is a transition to a frozen state. Sr Cr x Ga 12-x O 19 : Frozen spin state observed experimentally. Sr Cr x Ga 12-x O 19 : Frozen spin state observed experimentally.

8. 8 Experimental Background:Structural Transition Zn Cr 2 O 4 – unusual transition observed. Zn Cr 2 O 4 – unusual transition observed. Jahn-Teller transition to ordered (Neèl) state in pure system at T N <<|θ c-w | Jahn-Teller transition to ordered (Neèl) state in pure system at T N <<|θ c-w | Lattice tetragonally distorts, changing exchange interactions and relieving frustration. Lattice tetragonally distorts, changing exchange interactions and relieving frustration.

9. 9 Experimental Background: Random Strains Doping Cd for Zn – both non-magnetic ions – destroys all LRO for 3% doping. Doping Cd for Zn – both non-magnetic ions – destroys all LRO for 3% doping. Cd has ionic radius 1.3 times greater that of Zn. Cd has ionic radius 1.3 times greater that of Zn. Doping Ga for Cr - non-magnetic for magnetic doping - stable Neel state up to 25% doping. Expect to have much larger effect Doping Ga for Cr - non-magnetic for magnetic doping - stable Neel state up to 25% doping. Expect to have much larger effect

10. 10 Questions: 1) 1)Why are the Nèel States in ZnCr 2 O 4 so sensitive to doping on sites occupied by non- magnetic ions? 2) 2)Can random strains induce spin freezing and hence explain the observed spin-freezing transitions in GFAFMs generally?

11. 11 Tetrahedron with Random Strains Spin stiffness: free energy cost of twisting spins at a boundary – analogous to superfluidity order parameter. Spin stiffness: free energy cost of twisting spins at a boundary – analogous to superfluidity order parameter. Pure frustrated system: no stiffness Pure frustrated system: no stiffness Random strains: J → J+δJ ij Random strains: J → J+δJ ij δJ ij ’s induce stiffness δJ ij ’s induce stiffness Stiffness expected to scale as Stiffness expected to scale as Calculate stiffness on single tetrahedron. Calculate stiffness on single tetrahedron. Averaging over disorder Averaging over disorder where A > 0 and it is proportional to |δJ ij | θ

12. 12 Is this spin freezing different from what happens in a conventional spin glass? Conventional spin glasses: δJ ~ J Conventional spin glasses: δJ ~ J Strained GFAFMs: δJ << J Strained GFAFMs: δJ << J Strained GFAFMs: freezing is within ground state manifold of clean system. Strained GFAFMs: freezing is within ground state manifold of clean system. still obeyed. still obeyed. Local strain randomness selects preferred local flux directions. Local strain randomness selects preferred local flux directions.

13. 13 Projection of δJ ij onto Ground State Manifold At low T, can project δJ ij onto the flat bands. At low T, can project δJ ij onto the flat bands. The projection operator, for large r, can be calculated exactly: The projection operator, for large r, can be calculated exactly: P ij (r, r’) = 8π 2 A ij,lm (|r-r’| 2 δ l,m – 3x l x m )/|r-r’| 5 Novel aspect: Long-range – dipolar form for d=3. Novel aspect: Long-range – dipolar form for d=3. In clean system can calculate the eigenvalues of the interaction matrix. In clean system can calculate the eigenvalues of the interaction matrix. Two eigenvalues are k- independent: flat bands. Two eigenvalues are k- independent: flat bands.

14. 14 Monte Carlo Simulations: Heat Capacity MC simulations on pyrochlore with antiferromagnetic Heisenberg spins. MC simulations on pyrochlore with antiferromagnetic Heisenberg spins. δJ ij uniformly distributed in [ –δJ, δJ ] δJ ij uniformly distributed in [ –δJ, δJ ] Heat Capacity without strain per spin: 3/4 k B Heat Capacity without strain per spin: 3/4 k B With strain: (1-1/N)k B With strain: (1-1/N)k B Apparently no singularity in heat capacity with strain: consistent with spin-glass-like transition. Apparently no singularity in heat capacity with strain: consistent with spin-glass-like transition.

15. 15 Heat Capacity

16. 16 Monte Carlo Simulations: Frozen State Measure correlation function Measure correlation function where runs 1 and 2 on same strained lattice but different starting spin configurations. Develops long range order below temperature, T f Develops long range order below temperature, T f T f proportional to δJ T f proportional to δJ The strains are selecting a unique subset of the ground state manifold The strains are selecting a unique subset of the ground state manifold

17. 17 Spin Freezing

18. 18 MC Simulations: Work in Progress Equilibration a problem at low T: At large time, Equilibration a problem at low T: At large time, does not converge to value of.. Next stage: use parallel tempering to cool down system. Next stage: use parallel tempering to cool down system. Qualitative confidence in results but need better cooling regime to get reliable quantitative results. Qualitative confidence in results but need better cooling regime to get reliable quantitative results.

19. 19 Conclusions The high sensitivity of ZnCrO to non-magnetic disorder shows importance of strains and magneto-elastic coupling. The high sensitivity of ZnCrO to non-magnetic disorder shows importance of strains and magneto-elastic coupling. Theoretical work and numerical simulations suggest that strains can cause the spin freezing transition. Theoretical work and numerical simulations suggest that strains can cause the spin freezing transition. Most frustrated systems are found to undergo a freezing transition at low enough temperature – our results are consistent with this. Most frustrated systems are found to undergo a freezing transition at low enough temperature – our results are consistent with this.

20. 20 Figure 8: Scaling of the critical temperature for spin-freezing as a function of disorder magnitude.

21. 21