Frustrated magnetism in 2D Collin Broholm Johns Hopkins University & NIST  Introduction Two types of antiferromagnets Experimental tools  Frustrated dimers in 2D Low T spectrum Field driven critical response H-T Phase diagram  Conclusions and plans

1/28/05PCCM SCES workshop2 Spin Versus Dimer Order 3D bi-partite AFMInteracting spin pairs 2×(2S+1)

1/28/05PCCM SCES workshop3 Conceptual Phase Diagram for Quantum Magnets T/J 1/S, frustration, 1/z, H, P, x, … Quantum Critical Heavy Fermion behavior High T C Superconductivity Non-fermi-liquids Chakravarty, Halperin, Nelson Sachdev

1/28/05PCCM SCES workshop4 1. Assume Neel order, derive spin wave dispersion relation 2. Calculate the reduction in staggered magnetization due to quantum fluctuations 3. If then Neel order is untenable diverges if through a surface Soft points in D=1 magnets Soft lines in D=2 magnets Soft planes in D=3 magnets Approach QCP from Neel state

1/28/05PCCM SCES workshop5 Geometrical Interpretation Weak connectivity: Order in one part of lattice does not constrain surroundings This can occur in higher dimensions when there is frustration and/or low coordination number, z.

1/28/05PCCM SCES workshop6 Why 2D frustration is particularly interesting  More interesting than 1D less complicated than 3D (at least for experimentalist)  Two being the lower critical dimension for Neel order, frustration can have a qualitative impact  High T C demands better understanding of 2D cooperative phenomena  The panoply of ordered states in frustrated systems offers potentials for applications

1/28/05PCCM SCES workshop7 Inelastic Neutron Scattering  We can measure dispersion relations  We determine structure through transition rate

1/28/05PCCM SCES workshop8 MACS spectrometer under construction at NIST MACS spectrometer under construction at NIST pipi pfpf ħQħQ

1/28/05PCCM SCES workshop9 Renormalized Classical 2D AFM Coldea et al. PRL (2001) La 2 CuO 4 Greven et al PRL (1994) Sr 2 CuO 2 Cl 2 J. Tranquada

1/28/05PCCM SCES workshop10 AFM on 2D kagome’ sandwich I. S. Hagemann et al. PRL (2001) T C /  CW << 1 ⇒ Near Quantum Critical

1/28/05PCCM SCES workshop11 DCS/NIST Satisfied Simplexes Gasparovic et al. (2005)

1/28/05PCCM SCES workshop12 Relieving Frustration in a tetrahedron Available from for $700

1/28/05PCCM SCES workshop13 Slowing local spin fluctuations at QCP DCS/NIST SCGO QS-Ferrite

1/28/05PCCM SCES workshop14 Anomalous Freezing with minimal disorder Order-parameter-like Development of Small “frozen” moment Order-parameter-like Development of Small “frozen” moment Low T specific heat indicates spin-wave-like normal modes Low T specific heat indicates spin-wave-like normal modes IRIS/ISIS A “simplex glass”

1/28/05PCCM SCES workshop15 Frustrated Magnet with spin gap   (meV) Stone et al. PRB (2001) (C 4 H 12 N 2 )Cu 2 Cl 6 (PHCC)

1/28/05PCCM SCES workshop16 2D dispersion and first moment   (meV) h

1/28/05PCCM SCES workshop17 Frustrated bonds in PHCC Green colored bonds increase ground state energy The corresponding interactions are frustrated Green colored bonds increase ground state energy The corresponding interactions are frustrated

1/28/05PCCM SCES workshop18 Multi-magnon excitations

1/28/05PCCM SCES workshop19 Bound state meets continuum

1/28/05PCCM SCES workshop20 Bound state meets continuum

1/28/05PCCM SCES workshop21 Field driven QCPField driven level Crossing 0 1 H J H

Dynamic correlations at the QCP h (r. l. u.) T=8 K T=3 K T=0.04 K  Fix field at H C =7.5 T  Heating at the QCP yields finite ħ maximum  This indicates triplon repulsion

1/28/05PCCM SCES workshop23 Theory of dynamic correlations at QCP Subir Sachdev, Yale and E. R. Dunkel, Harvard  from self consistent Hartree Fock U =5 meV 2  ~ fitted at each T overall scale fitted at each T

T=8 K T=3 K Dynamic correlations at the QCP h (r. l. u.) T=8 K T=3 K T=0.04 K

1/28/05PCCM SCES workshop25 Non-linear magnetization Stone et al to (2005)

1/28/05PCCM SCES workshop26 Phase diagram for frustrated 2D magnet Stone et al to (2005)

1/28/05PCCM SCES workshop27 Cross over to 2D gapless phase w/o LRO Stone et al to (2005)

1/28/05PCCM SCES workshop28 Phase diagram for frustrated 2D magnet Stone et al to (2005)

1/28/05PCCM SCES workshop29 Evidence of non-monotonic H C (T) Stone et al to (2005)

1/28/05PCCM SCES workshop30 Anomalies near QCP Scenarios:  Cross over between 2 regimes: — 3D ordering of magnetized dimers at low T — 3D ordering of 2D bose condensate at higher T  Additional low T degrees of freedom — Nuclear spin order — Spin Peierls like lattice distortion Stone et al to (2005) Neel order BEC Singlet

1/28/05PCCM SCES workshop31 Summary  Three regimes for quasi-2D magnets Renormalized classical (La 2 CuO 4 ) Near quantum critical and gapless (QS-ferrite) Isolated singlet ground state systems (PHCC)  PHCC: Excitation spectrum Evidence for competing interactions Complex cooperative singlet Bound state decay into continuum  PHCC: phase diagram 3D Neel phase within bose condensed triplon phase Cross over and/or extra degrees of freedom at QCP

1/28/05PCCM SCES workshop32 Outlook on 2D frustration  Excitations in gapless 2D phase  More complete clarification of QCP anomalies to distinguish explanations Cross over between different regimes for interlayer coupling Novel degrees of freedom, a generalized spin Peierls effect?  Quantum impurities in PHCC  Novel neutron instrumentation Access to spin dynamics in small samples Comprehensive surveys of Q-E space Parametric studies versus H and pressure

1/28/05PCCM SCES workshop33 Collaborators PHCC M. Bouloubasis N. Harrison M. Kenzelmann D. H. Reich M. B. Stone H. Tao P. Vorserwisch I. Zaliznyak Pyrochlore slabs G. Aeppli R. Cava G. Gasparovic I. S. Hagemann S. H. Lee Theory E. R. Dunkel S. Sachdev