Joseph Snider * and Clare Yu University of California, Irvine

1. 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

2. 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]

3. 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)

4. 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.

5. Examples of Dipolar Glasses (that we will focus on) • Two level systems in glasses • LiHoxY1-xF4 (Holmium ions have Ising magnetic dipole moments)

6. 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.

7. 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 undergo a spin glass phase transition at low temperatures?

8. 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)

9. 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.

10. 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)

11. 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 • Ewald summation to handle long range interaction

12. 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

13. High T qEA=0 Low T qEA≠0 time time 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:

14. 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)

15. Order Parameter Distribution P(q,T) Low T High T • P(q) is Gaussian at high T • P(q) is bimodal at low T Concentration x = 4.5%, L = 10 (46 dipoles), T = 5, 1.6, 1.1, 0.9, 0.5 How do we determine if there is a transition?

16. 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 20% 4.5% 100%

17. Define Tg where P(q,T) is flattest L = 6, 8, 10, 12 • D(T) is deviation of P(q,T) from flatness (D is variance) • Tg at minimum of D(T) vs. T plot Tg Low T L = 4, 6, 8 High T

18. Dilute Dipole Glass Transition Temperature Vanishes as N→∞ This may explain why no dipole phase transition is observed. 100% Slope = -1/2 for x = 1%, 4.5%, 8%, 12%, 15.5%, and 20%

19. 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)

20. 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

21. 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 Extrapolate SN to N→∞ SN→∞ vs. T 100% 1/N=

22. 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.

23. Specific Heat CV Simulations x = 4.5% Experiment LiHoxY1-xF4 (Quilliam et al., 2007) • No sharp features • No indication of a phase • transition • Spin glass CV usually • a broad bump Simulations x = 20%

24. 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 S0 vs. Concentration Experiment Theory (Quilliam et al., PRL 98, 037203 (2007))

25. 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

26. 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.

27. 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)

28. 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)

29. The End

30. 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=0.045 x=0.12 x=0.20