Frustrated diamond lattice antiferromagnets

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).

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

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

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

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

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

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 1 5 10 0 900 c MnAl O 4 CoAl O 4 Very limited theoretical understanding V. Fritsch et al. PRL 9, 116401 (004); N. Tristan et al. PRB 7, 174404 (005); T. Suzuki et al. (unpublished)

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 1 5 10 0 900 f Θ CW T c MnAl O 4 Very limited theoretical understanding CoAl O 4 s 3/ V. Fritsch et al. PRL 9, 116401 (004); N. Tristan et al. PRB 7, 174404 (005); T. Suzuki et al. (unpublished)

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

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

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)

T 0 physics: ground states H 1 i Si S + i S i S Neel 0 1/8 1 Useful rewrite of Hamiltonian: 4 3 0 1 H + [ S0 + ( S1 + S + S3 + S 1 4) / 4] t ( 1 /8) Si S i

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.

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

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

Aside on self-consistent approach - Expand Hamiltonian in fluctuations: δs i S i S i ; H Interaction terms H H + H3 + 4 +... -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)

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

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

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

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

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

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!

Structure factor in spiral spin liquid regime 1 MnSc S 4 0.85 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

Spin liquid correlations analytically Spherical model Describes spin liquids in kagome, pyrochlore antiferromagnets Predicts structure factor data collapse S 1 S N 1 0.85 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/

Spin liquid correlations analytically Spherical model Describes spin liquids in kagome, pyrochlore antiferromagnets Predicts structure factor data collapse S 1 S N 1 0.85 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/

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

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 / 1 0.85 Lowest-T order determined energetically, not entropically Experiment sees 110 order (favored by AFM 3 ) A. Krimmel et al. PRB 73, 014413 (006); M. Mucksch et al. (unpublished)

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, 014413 (006); M. Mucksch et al. (unpublished)

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, 174404 (005); A. Krimmel et al. Physica B 378-380, 583 (006); T. Suzuki et al. (unpublished)

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

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