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 LiHo x Y 1-x F 4 (Holmium ions have Ising magnetic dipole moments) Eu x Sr 1-x S (insulating spin glass) Ising rubies [(Cr x Al 1-x )O 3 ]

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/r 3 interactions undergo a finite temperature spin glass transition (Katzgraber, Young, Bray, and Moore). But 1/r 3 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 LiHo x Y 1-x F 4 (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 C V ~ T and thermal conductivity ~ T 2.

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 LiHo x Y 1-x F 4 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 LiHo x Y 1-x F 4, 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” T g ~ 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 (E i )/n (E f )] where E i =initial energy and E f = final energy Accepted flip: n(E f ) → γn(E f ) where γ > 1 Rejected flip: n(E i ) → γn(E i ) Want histogram of visited energies h(E) to be flat: h(E) > (ε ) where 0 < ε < 1 (typical ε ≈ 0.95) Once flat enough, set γ → √ γ, set h(E)=0, iterate 20 times

13. Historical Edwards-Anderson Order Parameter q EA for Spin Glasses At high temperatures a spin glass has random fluctuating spins S i so that = 0 At low temperatures a spin glass has frozen spins Edwards-Anderson order parameter: time Low T q EA ≠0 time High T q EA =0

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) 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?

16. Binder’s g doesn’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 T C, g scales as Used to find T C : Binder’s g vs. T curves for different size systems cross at T C 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 T g 100% 4.5% 20%

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

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

19. Comments on Absence of a Dipole Glass Transition Unexpected since 3D Ising spin glasses with 1/r 3 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 = S tot (T) = ln Z(T) + Ē/T Entropy/particle = S N (T) =S tot (T)/N S N→∞ (T→0, x) tends to increase as x decreases 100% Extrapolate S N to N→∞ S N→∞ vs. T 1/N=

22. Comments on Finite Low T Entropy Fit data to S(T, x) = AT λ + S N →∞ (T→0, x) S N→∞ (T→0, x) increases as the concentration x decreases below x = 20% Finite S N→∞ (T→0, x) indicates accessible low energy states Classical system does not violate 3 rd Law of Thermodynamics, e.g., noninteracting spins.

23. Specific Heat C V Simulations x = 4.5% Simulations x = 20% Experiment LiHo x Y 1-x F 4 (Quilliam et al., 2007) No sharp features No indication of a phase transition Spin glass C V usually a broad bump

24. Specific Heat Experiments on LiHo x Y 1-x F 4 No sharp features in specific heat Residual entropy S 0 increases as x decreases Experimental S 0 order of magnitude larger than theory S 0 > 0 implies no spin glass phase transition (Quilliam et al., PRL 98, 037203 (2007)) S 0 vs. Concentration Experiment Theory

25. Is LiHo x Y 1-x F 4 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 H t can be used to tune quantum phase transition. Thus, LiHo x Y 1-x F 4 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 H t ~ 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 H t destroys spin glass phase transition.

27. Magnetic Susceptibility Experiments M = χ 1 H + χ 3 H 3 + … χ 3 should diverge for a spin glass transition Fit to χ 3 ~ [(T-T g )/T] -γ gives unphysical values of parameters No phase transition for LiHo x Y 1-x F 4 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” T g ~ 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 S N→∞ (T→0, x) increases as the concentration x decreases below x = 20% Finite S N→∞ (T→0, x) indicates accessible low energy states Classical system does not violate 3 rd Law of Thermodynamics, e.g., noninteracting spins. x=0.045 x=0.12 x=0.20