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

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

3. 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

4. From H. Takagi
Checkerboard lattice

5. 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

6. 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.

7. 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!

8. 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

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

10. 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?

11. 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?

12. 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,

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

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

15. 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

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

17. 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

18. 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

19. 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)

20. 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

21. 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.

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

23. 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?

24. 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

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

26. 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

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

28. 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

29. 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

30. 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

31. 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)

32. Quantum Spin Liquids

33. 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

34. 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!

35. 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