Impurities in Frustrated Magnets Leon Balents, UCSB Disorder, Fluctuations, and Universality, 2008

Collaborators Doron Bergman (Yale) Jason Alicea (Caltech) Simon Trebst (MS Station Q) Lucile Savary (ENS Lyon)

What is frustration? Competing interactions Can’t satisfy all interactions simultaneously Optimization is “frustrating” “People need trouble – a little frustration to sharpen the spirit on, toughen it. Artists do; I don't mean you need to live in a rat hole or gutter, but you have to learn fortitude, endurance. Only vegetables are happy.” – William Faulkner

From H. Takagi Checkerboard lattice

Frustration: Constrained Degeneracy When kBT ¿ J, system (classically) is constrained to ground state manifold Triangular lattice Ising antiferromagnet One dissatisfied bond per triangle Entropy 0.34 kB / spin Pyrochlore Heisenberg antiferromagnet Pyrochlore “Spin ice”: 2 in/2 out Ising spins Pauling entropy ¼ ½ ln(3/2) kB / spin

Challenge: spin liquid regime Frustration leads to suppressed order “Frustration parameter” f=CW/TN & 5-10 System fluctuates between competing ordered states for TN<T<CW What is the nature of the correlated liquid? Spin liquid This is a general problem in correlated electrons. Order only appears much below the dominant energy scale (e.g. U, t for Hubbard model). How to describe the strongly correlated system “constrained” by these strong interactions (like t-J model is) but not yet ordered? Another similar problem: spin incoherent 1DEG.

Quantum Spin Liquids f = CW/TN =1 : quantum paramagnetism RVB and gauge theories… Many recent experimental candidates Herbertsmithite kagome Na4Ir3O8 hyperkagome NiGa2S4 triangular s=1 -(BEDT) organic triangular lattice FeSc2S4 diamond lattice spin-orbital liquid + + … Only one ¼ consistent with RVB/gauge theory!

One class: “dipolar” spin liquids Classical pyrochlore spin liquids are “emergent diamagnets” Local constraint: Dipolar correlations Basically same structure believed to hold for spin ice materials, and similar one for spinel chromites in half-polarized magnetization plateau region. Similar story for various spin Hamiltonians on “bi-simplex” lattices. Youngblood and Axe, 1980 Isakov, Moessner, Sondhi 2003 Y2Ru2O7: J. van Duijn et al, 2007

A Problem Signatures of spin liquid correlations in neutron scattering are subtle Not peaks Often single crystal neutron scattering in not available

Can impurities be clarifying? Impurities may induce observable distortions in the correlated medium C.f. Friedel oscillation Long-range impurity interactions? Can look for differences in impurity-induced glassy states Formation with even weak impurities? Unconventional properties and transitions?

Some experimental features Appearance of spin glass state for very weak disorder (e.g. in kagome and spin ice materials) Commonly observed T2 specific heat in glass state NiGa2S4 – coherent spin waves in disordered state, without observable glass transition MnSi – partial order: disordered spirals?

Strange spin glasses in HFMs SCGO: SrCr9pGa12-9pO19 s=3/2 kagome Tg independent of disorder at small dilution? Unusual T2 specific heat? nearly H-independent! Ramirez et al, 89-90.

NiGa2S4 : Spin-1 Triangular Lattice AFM Nakatsuji et al. Science (2005) CW=-80K

Spin freezing without C(T) anomaly Nakatsuji et al. (2005) Single crystal

Excitations from low T state C. Broholm “Early” dispersion relation What is clear so far: Spin wave like modes at low T A “slow” low E mode throughout zone + A highly dispersive mode

A-site spinel CoAl2O4 Structure factor consistent with frozen superposition of spirals Unusual T2.5 heat capacity

Back to the dipolar spin liquid… Ising Pyrochlore = dimer model Down spin = dimer Very generally, dimer models on bipartite lattices show dipolar phases at high temperature T¿ J: 3 up and 1 down spin T¿ J: 2 up and 2 down spins 1 “dimer” per diamond site 2 “dimers” per diamond site In a field

Dilution Replace magnetic atom by non-magnetic one In dimer picture, this removes a link on which a dimer may sit + - 2 un-satisfied tetrahedra Dipole source! Indeed observe long-range disturbance

Random bonds Jij ! Jij+Jij Degeneracy of different states obviously broken Expect: glassy state for kBT ¿ |Jij| Q: What is the nature of the glass transition? Numerical evidence of Saunders and Chalker for such behavior in classical Heisenberg pyrochlore (2007)

Expect unconventional transition General argument (Bergman et al, 2006): Spin glass order parameter does not describe the dipolar correlations in the paramagnetic phase Can be argued that transition should be described by a gauge theory in which the Higgs phenomena quenches the dipolar fluctuations in the low temperature state Holds for any interactions (also non-random) that quench the entropy Recent examples studied by Alet et al and Pickles et al

A simple and dramatic example Classical cubic dimer model Hamiltonian Model has unique ground state – no symmetry breaking. Nevertheless there is a continuous phase transition! - Without constraint there is only a crossover.

Numerics (courtesy S. Trebst) Specific heat T/V “Crossings”

Many open issues How do multiple non-magnetic impurities interact in a dipolar spin liquid? What is the phase diagram of a bond-diluted dimer model? Purely geometrical problem with no energy scale! What is the nature of the glass transition from a dipolar Ising spin liquid?

Other spin liquids? A-site spinels Many materials! s = 5/2 FeSc2S4 CoRh2O4 Co3O4 MnSc2S4 1 5 10 20 900 MnAl2O4 CoAl2O4 s = 3/2 Frustration is due to competing exchange interactions on a diamond lattice Creates very large degeneracy but smaller than in pyrochlores

Ground state evolution Coplanar spirals Evolving “spiral surface” Neel q 1/8 Spiral surfaces:

Monte Carlo: “order by disorder” Parallel Tempering Scheme Tc rapidly diminishes in Neel phase “Order-by-disorder”, with sharply reduced Tc “spiral spin liquid” Reentrant Neel

Effects of impurities? Competing tendencies Break spiral degeneracy: stabilize order? Locally random: create glass? What would Thomas do? Use scaling arguments!

Single impurity Q1: How does a single impurity affect the spiral degeneracy? A1: far from the impurity, the system must have a uniform spiral wavevector A2: for given impurity, there is a finite energy, a(k), which depends upon the wavevector at infinity A3: approach to the uniform state is power-law and anisotropic, but relatively fast Divergent energy

Multiple impurities In general, several types of impurities may be present In spinel, dominant impurity is “inversion” defect – magnetic atom on B site. There are four such defects not equivalent by translations. Each impurity has its own a(k) favoring different discrete k spiral states J2/J1=0.2: (111) favored

Multiple Impurities Q2: Does a low but non-zero density of impurities favor a disordered or ordered state? A: ordered Because of stiffness energy (k)2 L, wavevector tends to remain uniform over many impurities Average energy favors discrete ordered states J2/J1=0.2: <100> favored

Multiple Impurities Q2: Does a low but non-zero density of impurities favor a disordered or ordered state? A: ordered Because of stiffness energy (k)2 L, wavevector tends to remain uniform over many impurities Average energy favors discrete ordered states Fluctuations in impurity densities do not destroy order, like weak random fields in 3d Ising model (Imry-Ma)

Quantum Spin Liquids

Na3Ir4O7 Hyperkagome A quantum paramagnet: CW¼ -650K 5d5 LS Ir4+  » Const C » T2 inconsistent with quasiparticle picture? Same behavior in other s=1/2 materials! x = 0 0.01 T 0.1 T 1 T 5 T Tg  10-3 emu/mol Ir 10K

Dilution (Ti doping) releases spins Two population fit of  (P. Schiffer and I. Daruka PRB, 56, 13712(1997) ~ -4 K On the other hand, at lower temperature, susceptibility is widely deviating from Curie-Weiss law with Ti doping. This indicates nearly free spin with small Weiss temperature is added to correlated spin with large negative Weiss temperature with Ti doping. I tried to fit this susceptibility by two spin population model represented as this formula in accordance with Schiffer and Daruka. These solid line is results of this fit, you can see susceptibility is nicely fitted by this formula in all temperature range, where spin 1 is AF spin of almost Ir ions and spin 2 is nearly free spin induced by Ti doping. I show C2 or ratio of nearly free spin to all spins here, you can see nearly free spins are increasing with Ti doping. Approximately 0.3B released per Ti!

Conclusions Impurities can reveal the correlations in spin liquid states Experiments and theory point to new types of glassy phases and transitions in these materials Even for the best understood “dipolar” spin liquid, impurity physics is largely mysterious