Frustrated diamond lattice antiferromagnets

Transcription

1 Frustrated diamond lattice antiferromagnets ason Alicea (Caltech) Doron Bergman (Yale) Leon Balents (UCSB) Emanuel Gull (ETH Zurich) Simon Trebst (Station Q) Bergman et al., Nature Physics 3, 487 (007).

2 Outline Introduction Frustration, degeneracy, & emergent phenomena Diamond lattice antiferromagnets Overview of experiments Theory of frustrated diamond lattice antiferromagnets Ground states (highly degenerate spirals) Stability & order-by-disorder Monte Carlo simulations Spiral spin liquid Comparison to experiment Summary & future directions

3 Frustration, degeneracy, & emergent phenomena Frustration all interactions not satisfiable simultaneously General principle: The presence of many competing states often leads to interesting physics Quantum Hall effect High-T c superconductors Frustrated magnets (Mott insulators)? 1 Highly degenerate ground states (pyrochlore, kagome, FCC, etc.) High sensitivity to perturbations Spin-glass behavior Spin-liquid physics Order-by-disorder

4 Experimental signatures of frustration 0 At high temperatures, Curie-Weiss law holds. High-T paramagnet ΘCW 1 χ ; ΘCW ( Exchange ' s) Τ Θ CW T

5 Experimental signatures of frustration Long-range order 0 At high temperatures, Curie-Weiss law holds. High-T paramagnet ΘCW 1 χ ; ΘCW ( Exchange ' s) Τ Θ CW T At low temperatures, systems typically order. Useful diagnostic: frustration parameter f Θ CW Tc

6 Experimental signatures of frustration Long-range order 0 T c At high temperatures, Curie-Weiss law holds. Spin liquid High-T paramagnet ΘCW 1 χ ; ΘCW ( Exchange ' s) Τ Θ CW T At low temperatures, systems typically order. Useful diagnostic: frustration parameter Highly frustrated systems f > 5-10 f Θ CW Tc Low-T ordering - Key challenges: & mechanisms Characterizing spin liquid correlations

7 Frustrated diamond lattice antiferromagnets: Materials Many materials take on the normal spinel structure: AB X 4 A diamond lattice B pyrochlore X FCC Focus: spinels with magnetic A-sites (only) CoRh O 4 Co 3 O 4 MnSc S 4 FeSc S 4 f Θ CW T c MnAl O 4 CoAl O 4 Very limited theoretical understanding V. Fritsch et al. PRL 9, (004); N. Tristan et al. PRB 7, (005); T. Suzuki et al. (unpublished)

8 Frustrated diamond lattice antiferromagnets: Materials Many materials take on the normal spinel structure: AB X 4 A diamond lattice B pyrochlore X FCC Focus: spinels with magnetic A-sites (only) s CoRh O 4 Co 3 O 4 MnSc S 4 s 5/ FeSc S 4 Orbital degeneracy f Θ CW T c MnAl O 4 Very limited theoretical understanding CoAl O 4 s 3/ V. Fritsch et al. PRL 9, (004); N. Tristan et al. PRB 7, (005); T. Suzuki et al. (unpublished)

9 Frustration on the bipartite diamond lattice?? Naïve Hamiltonian: H 1 i S i S Classical spins (s 3/, 5/ for materials of interest) 1 Diamond lattice FCC sublattices coupled via 1

10 Frustration on the bipartite diamond lattice?? Naïve Hamiltonian: H 1 i S i S No Frustration

11 Frustration on the diamond lattice Remedy: nd neighbor exchange H 1 i Si S + i S i S Generates strong frustration 1 and expected to be comparable due to similarity in exchange paths W. L. Roth,. Phys. (Paris) 5, 507 (1964)

12 T 0 physics: ground states H 1 i Si S + i S i S Neel 0 1/8 1 Useful rewrite of Hamiltonian: H + [ S0 + ( S1 + S + S3 + S 1 4) / 4] t ( 1 /8) Si S i

13 T 0 physics: ground states H Neel 1 i Si S + i Highly degenerate coplanar spirals S 0 1/8 1 i S Direction & pitch of spirals characterized by a wavevector residing on a surface in momentum space! Spiral surfaces 1 0.

14 T 0 physics: ground states H Neel 1 i Si S + i Highly degenerate coplanar spirals S 0 1/8 1 i S Direction & pitch of spirals characterized by a wavevector residing on a surface in momentum space! Spiral surfaces

15 T 0 physics: ground states H Neel 1 i Si S + i Highly degenerate coplanar spirals S 0 1/8 1 i S Direction & pitch of spirals characterized by a wavevector residing on a surface in momentum space! Spiral surfaces

16 T 0 physics: ground states H Neel 1 i Si S + i Highly degenerate coplanar spirals S 0 1/8 1 i S Direction & pitch of spirals characterized by a wavevector residing on a surface in momentum space! Spiral surfaces

17 Low-T physics: Can long-range order occur? Stability nontrivial due to massive spiral degeneracy. - Expand Hamiltonian in fluctuations: δs i S i S i Arbitrary ground state order -At T 0, branch of normal modes has infinite # of zeros! ω0 ( q) 0 For all q on surface Naively, fluctuations diverge δs i 3 d q ω ( q) ~ T 0

18 Key ideas: Order-by-disorder stabilization Only symmetry-required zeros in ω 0 (q) are the Goldstone modes Thermal fluctuations lifts the remaining accidental zeros entropy stabilizes long-range order! Needed: finite-t corrections to ω 0 (q) on spiral surface Perturbation theory insufficient Use self-consistent approach instead / 3 Answer: ω T ( q) ω ( q) + T Σ( q) 0 3 ~ d q δs T ~ ( ) T i ω q T 1/ 3 (Fluctuations small at low T) Non-analytic T-dependence unconventional thermodynamic behavior, e.g., C v A + BT 1/ 3

19 Aside on self-consistent approach - Expand Hamiltonian in fluctuations: δs i S i S i ; H Interaction terms H H + H Get self-energy self-consistently for divergent mode Σ (q) +... Full propagator + Σ ( q) T Γ ( q, k) G( k); k 1 G( k) ω 0 ( k) + Σ( k) α -For q on surface, assume Σ( q) ~ T Σ( q) ω T ( q) ω ( q) + T 0 / 3 Σ( q)

20 Order-by-disorder selection Long-range order occurs but which state does entropy select? Need Free Energy for all Q on spiral surface F( Q) E TS( Q) Entropy favors states with highest density of nearby low-energy states Complex phase structure emerges: 1/8 1/4 ~1/ ~/3 Green Free energy minima, red low, blue high 1

21 Order-by-disorder selection Long-range order occurs but which state does entropy select? Need Free Energy for all Q on spiral surface F( Q) E TS( Q) Entropy favors states with highest density of nearby low-energy states Complex phase structure emerges: 1/8 1/4 ~1/ ~/3 Green Free energy minima, red low, blue high 1

22 Order-by-disorder selection Long-range order occurs but which state does entropy select? Need Free Energy for all Q on spiral surface F( Q) E TS( Q) Entropy favors states with highest density of nearby low-energy states Complex phase structure emerges: 1/8 1/4 ~1/ ~/3 Green Free energy minima, red low, blue high 1

23 Order-by-disorder selection Long-range order occurs but which state does entropy select? Need Free Energy for all Q on spiral surface F( Q) E TS( Q) Entropy favors states with highest density of nearby low-energy states Complex phase structure emerges: 1/8 1/4 ~1/ ~/3 Green Free energy minima, red low, blue high 1

24 Order-by-disorder selection Long-range order occurs but which state does entropy select? Need Free Energy for all Q on spiral surface F( Q) E TS( Q) Entropy favors states with highest density of nearby low-energy states Complex phase structure emerges: 1/8 1/4 ~1/ ~/3 Green Free energy minima, red low, blue high 1

25 Monte Carlo simulations Parallel tempering algorithm employed to dramatically improve thermal equilibration T c rapidly diminishes in Neel phase

26 Monte Carlo simulations Parallel tempering algorithm employed to dramatically improve thermal equilibration T c rapidly diminishes in Neel phase Spiral surface develops

27 Monte Carlo simulations Parallel tempering algorithm employed to dramatically improve thermal equilibration T c rapidly diminishes in Neel phase Reentrant Neel Spiral surface develops

28 Monte Carlo simulations Parallel tempering algorithm employed to dramatically improve thermal equilibration T c rapidly diminishes in Neel phase Order-by-disorder, with sharply reduced T c Reentrant Neel Spiral surface develops

29 Spin liquid physics Order-by-disorder occurs at low temperatures Broad spin liquid regime emerges due to low T c Can probe this physics experimentally via neutron scattering Order by disorder Spin liquid High-T paramagnet 0 Tc ΘCW T Spin structure factor directly images spiral surface & entropic free energy corrections!

30 Structure factor in spiral spin liquid regime 1 MnSc S Analytic free energy Numerical structure factor Free energy corrections visible for T c <T< 1.3 T c Spiral surface more robust: persists for T c <T< 3 T c Order by disorder Spiral spin liquid Physics dominated by spiral ground states 0 Tc 1.3T c 3T c T

31 Spin liquid correlations analytically Spherical model Describes spin liquids in kagome, pyrochlore antiferromagnets Predicts structure factor data collapse S 1 S N Peaked near surface # of sites MnSc S 4 Structure factor for one FCC sublattice Λ( q) cos qx 4 cos q 4 cos q 4 y z q 4 q 4 x y z + sin sin q 4 sin 1/

32 Spin liquid correlations analytically Spherical model Describes spin liquids in kagome, pyrochlore antiferromagnets Predicts structure factor data collapse S 1 S N Peaked near surface # of sites MnSc S 4 Structure factor for one FCC sublattice Quantitative agreement! (except very near T c ) Nontrivial experimental test, but need single crystals Λ( q) cos qx 4 cos q 4 cos q 4 y z q 4 q 4 x y z + sin sin q 4 sin 1/

33 What can we expect for experiments? Realistic Hamiltonian: H i + 1 S S S i i i S + δh Entropic free energy corrections vanish as T 0 Degeneracybreaking perturbations Energetic corrections from δh inevitably dominate at lowest T If δh small enough, expect order-by-disorder phase to appear at higher T

34 Comparison with experiment: MnSc S 4 q π (3/4,3/4,0) Spiral order Spiral order + Q diff π diffuse scattering Spin liquid with Q diff π diffuse scattering High-T paramagnet 0 1.9K.3K CW Θ T Entropy favors ~100 order Theoretical implications 1 is ferromagnetic here / Lowest-T order determined energetically, not entropically Experiment sees 110 order (favored by AFM 3 ) A. Krimmel et al. PRB 73, (006); M. Mucksch et al. (unpublished)

35 Comparison with experiment: MnSc S 4 (cont d) q π (3/4,3/4,0) Spiral order Spiral order + Q diff π diffuse scattering Spin liquid with Q diff π diffuse scattering 0 1.9K.3K CW Θ High-T paramagnet T Experiment Theory 3 1 /0 Intensity shifts from q to spiral surface as T washes out 3 Consistent with spiral spin liquid A. Krimmel et al. PRB 73, (006); M. Mucksch et al. (unpublished)

36 Much less known here Comparison with experiment: CoAl O 4 Strong frustration, sample dependent No sharp transition observed yet Powder neutron data + frustration suggest / 1 1/8 for this material CoAl O 4 MnSc S 4 N. Tristan et al. PRB 7, (005); A. Krimmel et al. Physica B , 583 (006); T. Suzuki et al. (unpublished)

37 Summary Many spinels constitute frustrated diamond lattice antiferromagnets MnSc S 4, CoAl O 4, etc. Simple 1 - model captures essential physics Continuous spiral ground state degeneracy Important ordering mechanism is order-by-disorder Spin correlations in spiral spin liquid reveals surface + entropic effects Theoretical predictions consistent with existing experiments

38 Single crystals wanted Future Directions Allow for more direct comparison Concrete experimental realization of order-by-disorder?? Explore spin dynamics for inelastic neutron scattering? Effects of disorder? Details of low-t order in MnSc S 4? Commensurate lock-in? Physics of spin + orbitally frustrated FeSc S 4? Exotic quantum ground state? Acknowledgments Lucile Savary Matthew Fisher Ryuichi Shindou Zhenghan Wang Alexander Krimmel Michael Muecksch Tomoyuki Suzuki