Dipole Glasses Are Different from Spin Glasses: Absence of a Dipole Glass Transition for

Randomly Dilute Classical Ising Dipoles

Joseph Snider* and Clare Yu

University of California, Irvine

*Now at Salk Institute, La Jolla, CA

Examples of Dipolar Glasses

• Electric dipole impurities in alkali halides• Dilute ferroelectrics• Frozen (magnetic) ferrofluids• Disordered magnets• Two level systems in glasses• LiHoxY1-xF4 (Holmium ions have Ising

magnetic dipole moments)• EuxSr1-xS (insulating spin glass)• Ising rubies [(CrxAl1-x)O3]

Do dilute Ising dipoles, randomly placed, undergo a classical spin

glass or dipole glass phase transition as the system is cooled?

Answer: No

(if the concentration is low enough)

Reason to Expect a Spin Glass Phase Transition for Dilute Dipoles

• 3D Ising spin glasses with 1/r3 interactions undergo a finite temperature spin glass transition (Katzgraber, Young, Bray, and Moore).

But• 1/r3 interactions are different from dipolar interactions.• Theoretical spin glasses are not dilute because they have a spin on every site.

Examples of Dipolar Glasses (that we will focus on)

• Two level systems in glasses

• LiHoxY1-xF4 (Holmium ions have Ising magnetic dipole moments)

Two Level Systems (TLS)• Two level systems are present in amorphous materials.• The microscopic nature of two level systems in glasses

is a mystery. • But one can think of an atom or a group of atoms that

can sit equally well in one of two positions.• Two level systems are responsible for the low

temperature properties of glasses such as specific heat CV ~ T and thermal conductivity ~ T2.

Two level systems (TLS) in glasses interact with one

another via:

• Electric dipole-dipole interactions (some TLS have electric dipole moments)

• Elastic strain field (stress tensor generalization of vector dipole interaction)

Question: Do two level systems undergoa spin glass phase transition at low temperatures?

Experimental Hint of a Spin Glass Transition for Dilute Dipoles

• Experimental hint of a transition: Change in slope of the dielectric constant (Strehlow, Enss, Hunklinger, PRL 1998)

Reasons why there may be no dipole glass transition

• TLS are very dilute (~ 100 ppm)

• Experiments do not see a transition in LiHoxY1-xF4 for x = 4.5%. (Holmium ions have Ising magnetic dipole moments.) Absence of a transition has been attributed to quantum mechanical effects.

• Need calculation of dilute classical dipolar system LiHoxY1-xF4, x=4.5%

Ghosh et al.

0.751/T

Conclusions of our Monte Carlo simulations of dilute Ising dipoles in

3D:• No phase transition for x = 1%, 4.5%, 8%,

12%, 15.5%, 20% which is consistent with experiments.

• Characteristic “glass transition temperature” Tg ~ 1/√N → 0 as N →∞ where N is the number of dipoles.

• Low temperature entropy per particle larger for lower concentrations.

Reference: J. Snider and C. Yu, PRB 72, 214203 (2005)

Monte Carlo simulations of dilute Ising dipoles in 3D

• Ising dipoles randomly placed on a simple cubic lattice

• Concentrations of x = 1%, 4.5%, 8%, 12%, 15.5%, 20%

• Dipole-dipole interaction

1 2 12 1 12 21 2 3

12

ˆ ˆ3( , )

p p r p r pH p p

r

• Ewald summation to handle long range interaction

Wang-Landau Monte Carlo

• Too difficult to equilibrate with traditional Monte Carlo• Wang-Landau Monte Carlo calculates density of states

n(E)• Can calculate temperature dependent quantities using

n(E)• Start with flat density of states (n(E) = 1)• Do random walk in energy space• Probability that state has energy E is product of probability of making a

transition to that state (~1/n(E)) times probability (~n(E)) that a state of energy E exists: H(E) ~ [1/n(E)] × [n(E)] = 1

• Single dipole flips accepted with probability =min[1, n (Ei)/n (Ef)] where Ei=initial energy and Ef = final energy

• Accepted flip: n(Ef) → γn(Ef) where γ > 1• Rejected flip: n(Ei) → γn(Ei)• Want histogram of visited energies h(E) to be flat: h(E) > (ε <h>) where 0 < ε

< 1 (typical ε ≈ 0.95)• Once flat enough, set γ → √ γ, set h(E)=0, iterate 20 times

Historical Edwards-Anderson Order Parameter qEA for Spin Glasses

• At high temperatures a spin glass has random fluctuating spins Si so that <Si> = 0

• At low temperatures a spin glass has frozen spins

• Edwards-Anderson order parameter:

lim ( ) (0)EA i it

q S t S

time

Low TqEA≠0

time

High TqEA=0

Generalized Edwards-Anderson Order Parameter q

• is dipole in state of current system

• is dipole in low energy state found before

• Note: Frozen system with nondegenerate ground state has perfect overlap q=1

• Find distribution P(q,E) from simulations

• Calculate P(q,T)

1 g si i

i

q p pN

sip

gip

/( , ) ( ) ( , ) E kT

E

CP q T n E P q E e

Z

Order Parameter Distribution P(q,T)

Concentration x = 4.5%, L = 10 (46 dipoles), T = 5, 1.6, 1.1, 0.9, 0.5

• P(q) is Gaussian at high T• P(q) is bimodal at low T

High T

Low T

How do we determine if there is a transition?

Binder’s gdoesn’t work

• Non-Gaussianity parameter g = 0 if P(q) is Gaussian (high T)• g = 1 if system is frozen, q = ± 1, and P(q) is bimodal (low T) • Near TC, g scales as • Used to find TC : Binder’s g vs. T curves for different size systems

cross at TC if there is a second order phase transition• Binder’s g curves cross for 100%, but not for x ≤ 20%• Need another way to find Tg

4

22

13 where ( , )

2m m

q

qg q q P q T

q

100%

4.5% 20%

1/~ ( )Cg g L T T

Define Tg where P(q,T) is flattest• D(T) is deviation of P(q,T)

from flatness (D is variance)

• Tg at minimum of D(T) vs. T plot

23( ) ( , ) ( , )

qq

D T L P q T P q T

L = 6, 8, 10, 12

L = 4, 6, 8High T

Low T

Tg

Dilute Dipole Glass Transition Temperature Vanishes as N→∞

10 as number of dipoles gT N

N

100%

for x = 1%, 4.5%, 8%, 12%, 15.5%, and 20%

This may explain whyno dipole phase transition is observed.

Slope = -1/2

Comments on Absence of a Dipole Glass Transition• Unexpected since 3D Ising spin glasses with 1/r3 interactions

have a transition• Dilute dipoles: P(q) is flat as N → ∞ and T → 0 • Spin glasses: P(q) is bimodal as N → ∞ and T → 0 • Model spin glasses have every site occupied so nearby spins

have stronger interactions than distant spins and produce large barriers between “ground state” configurations

• Dilute dipolar system has empty nearby sites so low energy configurations are determined by weakly interacting distant dipoles that produce low energy barriers between “ground state”configurations

• May explain absence of TLS phase transition

X = 4.5%, L=10 (46 dipoles)

Caveat: Absence of dilute dipolar transition may explain absence of TLS dipolar

transition, BUT• TLS are different from dipoles• TLS have energy asymmetry analogous to random local

field which tends to destroy phase transitons• TLS are not uniaxial (Ising) dipoles; rather they can

point in any direction• TLS are stress tensors that can interact via the strain field

with an interaction analogous to that of vector dipoles• Experimentally seen transition in dielectric constant may

not involve dipoles or TLS

Finite Low Temperature Entropy• Total Entropy = Stot(T) = ln Z(T) + Ē/T

• Entropy/particle = SN (T) =Stot(T)/N

• SN→∞ (T→0, x) tends to increase as x decreases

100%

Extrapolate SN to N→∞ SN→∞ vs. T

1/N=

Comments on Finite Low T Entropy• Fit data to S(T, x) = ATλ + SN →∞(T→0, x) • SN→∞ (T→0, x) increases as the concentration x decreases below x

= 20%• Finite SN→∞ (T→0, x) indicates accessible low energy states• Classical system does not violate 3rd Law of Thermodynamics, e.g.,

noninteracting spins.

Specific Heat CV

Simulationsx = 4.5%

Simulationsx = 20%

ExperimentLiHoxY1-xF4

(Quilliam et al., 2007)

• No sharp features• No indication of a phase transition• Spin glass CV usually a broad bump

Specific Heat Experiments on LiHoxY1-xF4• No sharp features in specific heat• Residual entropy S0 increases as x decreases• Experimental S0 order of magnitude larger than theory • S0 > 0 implies no spin glass phase transition

(Quilliam et al., PRL 98, 037203 (2007))

S0 vs. Concentration

Experiment

Theory

Is LiHoxY1-xF4 a Quantum Spin Glass?

• Experiments by Rosenbaum group led them to claim that x = 16.7% is a spin glass (Reich et al.).

• For x = 4.5%, they attribute lack of spin glass transition to quantum fluctuations (spin liquid or antiglass phase)

• They claim that transverse magnetic field Ht can be used to tune quantum phase transition.

• Thus, LiHoxY1-xF4 is considered a quantum spin glass

Naysayers (besides us): Other Theoretical Work

• Schechter and Stamp (2005): Hyperfine interactions in Ho are important. Transverse field Ht ~ tesla needed to see quantum fluctuations of Ising spins.

• Schecter and Laflorencie (2006); and Tabei et al. (2006): Assume spin glass ground state. Showed transverse field Ht destroys spin glass phase transition.

Magnetic Susceptibility Experiments• M = χ1H + χ3H3 + …• χ3 should diverge for a spin glass transition

• Fit to χ3 ~ [(T-Tg)/T]-γ gives unphysical values of parameters• No phase transition for LiHoxY1-xF4 with x = 16.5% and x = 4.5%

(Jönsson et al., 2007)

Conclusions of our Monte Carlo simulations of dilute Ising dipoles in 3D:• No phase transition for x = 1%, 4.5%, 8%, 12%, 15.5%,

and 20%.• Characteristic “glass transition temperature” Tg ~ 1/√N

→ 0 as N →∞ where N is the number of dipoles.• P(q) becomes flat as T→0 and N→∞.• Finite low temperature entropy per particle larger for

lower concentrations.• Lots of accessible low energy nearly degenerate states.• Lack of transition and residual entropy confirmed by

experiments.

Reference: J. Snider and C. Yu, PRB 72, 214203 (2005)

The End

Comments on Finite Low T Entropy• SN→∞ (T→0, x) increases as the concentration x decreases below x

= 20%• Finite SN→∞ (T→0, x) indicates accessible low energy states• Classical system does not violate 3rd Law of Thermodynamics, e.g.,

noninteracting spins.

x=0.045x=0.12x=0.20