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

Checkerboard lattice

From H. Takagi

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

Only one ¼ consistent with RVB/gauge theory!

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

+ + …

One class: “dipolar” spin liquids Classical pyrochlore spin liquids are

“emergent diamagnets” Local constraint: Dipolar correlations

Youngblood and Axe, 1980Isakov, 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

Single crystal

Nakatsuji et al. (2005)

Excitations from low T state“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

C. Broholm

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: 2 up and 2 down spins

2 “dimers” per diamond site

T¿ J: 3 up and 1 down spin

1 “dimer” per diamond siteIn 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

C

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!

1 900

FeSc2S4

10 205

CoAl2O4

MnSc2S4

MnAl2O4

CoRh2O4 Co3O4

s = 5/2

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

q0 12 JJ1/8

NeelEvolving “spiral surface”

85.012 JJ 2012 JJ4.012 JJ2.012 JJ

Spiral surfaces:

Monte Carlo: “order by disorder” Parallel Tempering Scheme

Tc rapidly diminishes

in Neel phase

“Order-by-disorder”,

with sharply reduced Tc

Reentrant Neel

“spiral spin liquid”

Effects of impurities? Competing tendencies

Break spiral degeneracy: stabilize order? Locally random: create glass?

What would Thomas do?

What would Thomas do? Use scaling

arguments!

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 -650K2500

2000

1500

1000

500

0

1 (

mol

Ir/

cm3 )

3002001000T (K)

Na4Ir3O8

H = 1 T

5d5 LS

Ir4+

S = 1/2

60

40

20

0Cm

/T (

mJ/

Km

ol I

r+T

i)

200150100500T (K)

8

6

4

2

0

S m (

J/K

mol

Ir+

Ti)

1.8

1.6

1.4

(1

0-3em

u/m

ol I

r)

1086420T (K)

2.0

1.8

1.6

1.4

10-3

x = 0

0.01 T0.1 T1 T5 T

Tg

1

0-3

em

u/m

ol Ir

» Const

C » T2

inconsistent with quasiparticle picture?

Same behavior in other s=1/2 materials!

0 10K

(P. Schiffer and I. Daruka PRB, 56, 13712(1997)

~ -4 K

6

4

2

0

C2

(10-2

emu/

Km

ol I

r)

0.40.30.20.10.0x (Ti)

0.16

0.12

0.08

0.04

0.00

nearly free spin/all spin

2500

2000

1500

1000

500

0

1 (

mol

Ir/

cm3 )

3002001000T (K)

Na4(Ir1 xTix)3O8

H = 1 Tx = 0

x = 0.05

x = 0.1x = 0.2

x = 0.3

Dilution (Ti doping) releases spins Two population fit of

Approximately 0.3B 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