Phase Transitions in Spin Glasses

Transcription

1 Phase Transitions in Spin Glasses Peter Young Talk available at peter/talks/sinica.pdf Supported by the Hierarchical Systems Research Foundation. Recent collaborators: H. G. Katzgraber, L. W. Lee, J. Pixley, D. Larson, V. Martin-Mayor, A. Tarancon, L. A. Fernandez, S. Gaviro, Talk at Academia Sinica, Taipei, December 14, p.1

2 Overview Basic Introduction What is a spin glass? Why are they important? Why are Monte Carlo simulations for spin glasses hard? Try to answer two important questions concerning phase transitions in spin glasses: Is there a phase transition in an isotropic Heisenberg spin glass? Is there a transition in an Ising spin glass in a magnetic field (Almeida-Thouless line)? p.2

3 What is a spin glass? A system with disorder and frustration. or Most theory uses the simplest model with these ingredients: the Edwards-Anderson Model: H = i,j J ij S i S j i h i S i. Interactions are quenched and are random (have either sign). Take a Gaussian distribution: [J ij ] av = 0; [Jij 2 ]1/2 av = J (= 1) Spins, S i, fluctuate and have m components: m = 1 (Ising) m = 2 (XY) m = 3 (Heisenberg). p.3

4 Spin Glass Systems Metals: Diluted magnetic atoms, e.g. Mn, in non-magnetic metal, e.g. Cu. RKKY interaction: J ij cos(2k F R ij ) R 3 ij Random in magnitude and sign, which gives frustration. Note: Mn (S-state ion) has little anisotropy; Heisenberg spin glass. Important because relevant to other systems: Vortex glass transition in superconductors Optimization problems in computer science (including solving optimization problems on a quantum computer) Protein folding Error correcting codes Spin glasses have the advantages that: Can be probed experimentally in exquisite detail by applying (time-dependent) magnetic fields. There are simple models which can be simulated efficiently. p.4

5 Spin Glass Phase Transition Phase transition at T = T SG. For T < T SG the spin freeze in some random-looking orientation. As T T + SG, the correlation length ξ SG diverges. The correlation S i S j becomes significant for R ij < ξ SG, though the sign is random. A quantity which diverges is the spin glass susceptibility χ SG = 1 N [ S i S j 2 ] av, i,j (notice the square), which is accessible in simulations. It is also essentially the same as the non-linear susceptibiliity, χ nl, defined by m = χh χ nl h 3 + (m is magnetization, h is field), which can be measured experimentally. For the EA model T 3 χ nl = χ SG 2 3. p.5

6 Non-linear susceptibility Non-linear susceptibility χ nl, is defined by m = χh χ nl h 3 + Find: χ nl (T T SG ), with γ generally in the range ; i.e. there is a finite temperature spin glass transition. e.g. results of Omari et al, (1983) for CuMn 1%. p.6

7 Slow Dynamics (free) energy 0 Slow dynamics The dynamics is very slow at low T. System not in equilibrium due to complicated energy landscape: system trapped in one valley for long times. valley barrier E valley configuration Many interesting experiments on non-equilibrium effects (aging). Here concentrate on equilibrium phase transitions. p.7

8 Memory and Rejuvenation Data for χ for a spin glass. On cooling the system waited (aged) at 12K (the data decreases). On cooling further the data went back to the reference curve (rejuvenation). On warming the data shows at dip at 12K even there was no waiting there (memory effect). Data from Jonason et al. (1998). χ" (a.u.) Decreasing T Increasing T Reference ageing at T 1 =12 K CdCr 1.7 In 0.3 S Hz T (K) p.8

9 Theory Mean Field Theory (Edwards-Anderson, Sherrington-Kirkpatrick, Parisi). Exact solution of an infinite range (SK) model. Finite T SG. The Parisi solution is a tour-de-force. Has an infinite number of order parameters. p.9

10 Theory Mean Field Theory (Edwards-Anderson, Sherrington-Kirkpatrick, Parisi). Exact solution of an infinite range (SK) model. Finite T SG. The Parisi solution is a tour-de-force. Has an infinite number of order parameters. Short-range (EA) models. Simulations on Ising systems also indicate a finite T SG (see later) in d = 3. Vector spin glasses? (See later.) p.9

11 Theory Mean Field Theory (Edwards-Anderson, Sherrington-Kirkpatrick, Parisi). Exact solution of an infinite range (SK) model. Finite T SG. The Parisi solution is a tour-de-force. Has an infinite number of order parameters. Short-range (EA) models. Simulations on Ising systems also indicate a finite T SG (see later) in d = 3. Vector spin glasses? (See later.) Equilibrium state below T SG. Two main scenarios: Replica Symmetry Breaking Droplet picture (DP) (Fisher (RSB), (Parisi). and Huse, Bray and Moore, McMillan). Assume short-range is similar to infinite-range. There is an AT line. Focus on the geometrical aspects of the low-energy excitations. No AT line in any d. p.9

12 Is there an AT line? (Ising) In MFT there s a transition in a field for an Ising spin glass, the Almeida Thouless (AT) line, from a spin glass (divergent relaxation times, RSB) to a paramagnetic (finite relaxation times, replica symmetric ) phase. The AT line is a ergodic-non ergodic transition with no symmetry change. H (a) H (b) H AT PM PM SG SG T c T Does an AT line occur in short range systems? T c T RSB: yes (see (a)) DP: no (see (b)) Experiments (dynamics) (Uppsala group): no Theory: conflicting claims. p.10

13 Overview Basic Introduction What is a spin glass? Why are they important? Why are Monte Carlo simulations for spin glasses hard? Try to answer two important questions concerning phase transitions in spin glasses: Is there a phase transition in an isotropic Heisenberg spin glass? Is there a transition in an Ising spin glass in a magnetic field (Almeida-Thouless line)? p.11

14 Why is Monte Carlo Hard? Dynamics is very slow. System is trapped in valley separated by barriers. Have to repeat the simulations for very many samples. [ ] av. But is trivially parallelizable. p.12

15 Parallel Tempering Problem: Very slow Monte Carlo dynamics at low-t ; System trapped in a valley. Needs more energy to overcome barriers. This is achieved by parallel tempering (Hukushima and Nemoto): simulate copies at many different temperatures: T T T T T T n 2 n 1 n Lowest T : system would be trapped: Highest T : system has enough energy to fluctuate quickly over barriers. Perform global moves in which spin configurations at neighboring temperatures are swapped. Result: temperature of each copy performs a random walk between T 1 and T n. Advantage: Speeds up equilibration at low-t. T p.13

16 Equilibration Equilibration test (for Gaussian distribution) e.g. for Ising [q l 1] av = 2 z T [U] av, where U = 1 N J ij S i S j (energy) i,j q l = 1 S i S j 2 ( link overlap ), N b = Nz/2. N b i,j z is the no. of neighbors, and J = 1. [ ] av is an average over samples. Data for Ising spin glass (H. Katzgraber) [q l 1] av and (2/z)T[U] av approach a common value from opposite directions, and, once they agree with each other, the results don t change if N sweep is increased further. p.14

17 Overview Basic Introduction What is a spin glass? Why are they important? Why are Monte Carlo simulations for spin glasses hard? Try to answer two important questions concerning phase transitions in spin glasses: Is there a phase transition in a Heisenberg spin glass? Is there a transition in an Ising spin glass in a magnetic field (Almeida-Thouless line)? p.15

18 Finite size scaling Assumption: size dependence comes from the ratio L/ξ bulk where ξ bulk (T T SG ) ν is the bulk correlation length. In particular, the finite-size correlation length varies as ξ ( ) L L = X L 1/ν (T T SG ), since ξ L /L is dimensionless (and so has no power of L multiplying the scaling function X). Hence data for ξ L /L for different sizes should intersect at T SG and splay out below T SG. Let s first see how this works for the Ising SG... p.16

19 Results: Ising FSS of the correlation length for the Ising spin glass. (from Katzgraber, Körner and APY Phys. Rev. B 73, (2006).) Method first used for SG by Ballesteros et al. but for the ±J distribution. The clean intersections (corrections to FSS visible for L = 4) imply T SG Prevously Marinari et al. found T SG = 0.95 ±0.04 using a different analysis Compelling evidence for finite T c in Ising (best is Hasenbusch et al. (2008)) p.17

20 Chirality Unfrustrated: Thermally activated chiralities (vortices) drive the Kosterlitz-Thouless-Berezinskii transition in the 2d XY ferromagnet. Frustrated: Chiralities are quenched in by the disorder at low-t because the ground state is non-collinear. Define chirality by: (Kawamura) { 1 κ µ i = 2 2 l,m sgn(j lm) sin(θ l θ m ), S i+ˆµ S i S i ˆµ, XY (µ square) Heisenberg XY Heisenberg p.18

21 Motivation for vector model Old Monte Carlo for Heisenberg: T SG, if any, seems very low, probably zero. Kawamura: T SG = 0 but transition in the chiralities, T CG > 0. This implies spin chirality decoupling. Subsequently Kawamura suggests that T SG > 0 but T SG < T CG. But: Alternative possibility of a single transition proposed by Nakamura and Endoh, Lee and APY, Campos et al, Pixley and APY. Here: describe recent work on FSS of the correlation lengths of both spins and chiralities for the Heisenberg spin glass. Useful since this was the most successful approach for the Ising spin glass. treat spins and chiralities on equal footing. p.19

22 Heisenberg Spin Glass (Martin-Mayor, Tarancon, Fernandez, Gaviro and APY (unpublished)). Note: much larger sizes than for Ising (barriers smaller). Took 7.5 megahours of CPU time! Spins and chiralities behave very similarly (but not identically) for this range of sizes. Are there two very close but distinct transitions? Viet and Kawamura, L 32, claim T CG = 0.145, T SG = From our data, the difference seems less than this or zero. The p.20 apparent small difference in transition temperatures may be due to corrections to scaling.

23 Overview Basic Introduction What is a spin glass? Why are they important? Why are Monte Carlo simulations for spin glasses hard? Try to answer two important questions concerning phase transitions in spin glasses: Is there a phase transition in an isotropic Heisenberg spin glass? Is there a transition in an Ising SG in a magnetic field (AT line)? p.21

24 Is there an AT line? (Ising) In MFT there s a transition in a field for an Ising spin glass, the Almeida Thouless (AT) line, from a spin glass (divergent relaxation times, RSB) to a paramagnetic (finite relaxation times, replica symmetric ) phase. The AT line is a ergodic-non ergodic transition with no symmetry change. H (a) H (b) H AT PM PM SG SG T c T Does an AT line occur in short range systems? T c T RSB: yes (see (a)) DP: no (see (b)) Experiments (dynamics) (Uppsala group): no Theory: conflicting claims. p.22

25 Is there an AT Line? (ii) Experiments:, no static divergent quantity; (χ nl doesn t diverge in a field). Simulations: According to RSB, χ SG diverges in a field, where now χ SG (k) = 1 N [( S i S j S i S j ) 2 ] av e ik (R i R j ). i,j Hence can use FSS of ξ L /L in the simulations to see if there is an AT line ξ L behaves as for the zero field case; so we look for intersections. p.23

26 Long-Range model in d = 1 [J 2 ij ] av 1,. rij 2σ Use diluted (diluted) version: Leuzzi et al (2008). The probability of a non-zero bond falls off like 1/rij 2σ, but the strength bond does not fall off. Analogy between short-range model in d-dimensions and the 1-d long-range model d (SR) σ [1 d (LR)] SK MF 1/2 6 (= d u ) 2/3 2.5 (= d l ) 1 Non MF σ 0 1/2 2/3 1 Infinite range Finite T T = 0 SG SG p.24

27 Results: AT-line (i) FSS of the correlation length for the Ising spin glass in a (Gaussian random) field of H r = 0.1 with σ = 0.75 (non mean field region). Lack of intersections implies no AT line down to this value of H r. (If there is an AT line, it only occurs for extremely small fields.) Katzgraber, Larson and APY PRL, 102, (2009) Hence expect no AT line in the non mean field region, i.e. d < 6. However, for the same model, with same σ and H r, (but with ±1 bonds), Parisi et al., arxiv: , find intersections and hence claim an AT line. p.25

28 Results: AT-line (ii) FSS of the correlation length for the Ising spin glass in a (Gaussian random) field of H r = 0.1 with σ = 0.60 (mean field region). (Katzgraber, Larson and APY). The implies intersection there is an AT line in the mean field region p.26

29 Conclusions Spin glasses are related to a range of problems in science, and have the advantage that there are simple models which can be simulated, and experiments can probe them in exquisite detail since they couple to a magnetic field. p.27

30 Conclusions Spin glasses are related to a range of problems in science, and have the advantage that there are simple models which can be simulated, and experiments can probe them in exquisite detail since they couple to a magnetic field. Monte Carlo simulations are very useful for spin glasses. p.27

31 Conclusions Spin glasses are related to a range of problems in science, and have the advantage that there are simple models which can be simulated, and experiments can probe them in exquisite detail since they couple to a magnetic field. Monte Carlo simulations are very useful for spin glasses. In this talk I showed: p.27

32 Conclusions Spin glasses are related to a range of problems in science, and have the advantage that there are simple models which can be simulated, and experiments can probe them in exquisite detail since they couple to a magnetic field. Monte Carlo simulations are very useful for spin glasses. In this talk I showed: Finite temperature transition in 3-d Ising SG is well understood. p.27

33 Conclusions Spin glasses are related to a range of problems in science, and have the advantage that there are simple models which can be simulated, and experiments can probe them in exquisite detail since they couple to a magnetic field. Monte Carlo simulations are very useful for spin glasses. In this talk I showed: Finite temperature transition in 3-d Ising SG is well understood. There is a finite-temperature transition in the three-dimensional Heisenberg spin glass. The spin-glass and chiral-glass transition temperatures are very close and may well be equal. p.27

34 Conclusions Spin glasses are related to a range of problems in science, and have the advantage that there are simple models which can be simulated, and experiments can probe them in exquisite detail since they couple to a magnetic field. Monte Carlo simulations are very useful for spin glasses. In this talk I showed: Finite temperature transition in 3-d Ising SG is well understood. There is a finite-temperature transition in the three-dimensional Heisenberg spin glass. The spin-glass and chiral-glass transition temperatures are very close and may well be equal. There does not appear to be an Almeida-Thouless line in three dimensions, though it may occur for d greater than a critical value (perhaps 6). p.27

35 Conclusions Spin glasses are related to a range of problems in science, and have the advantage that there are simple models which can be simulated, and experiments can probe them in exquisite detail since they couple to a magnetic field. Monte Carlo simulations are very useful for spin glasses. In this talk I showed: Finite temperature transition in 3-d Ising SG is well understood. There is a finite-temperature transition in the three-dimensional Heisenberg spin glass. The spin-glass and chiral-glass transition temperatures are very close and may well be equal. There does not appear to be an Almeida-Thouless line in three dimensions, though it may occur for d greater than a critical value (perhaps 6). (The last two are not yet universally accepted.) p.27

36 Conclusions Spin glasses are related to a range of problems in science, and have the advantage that there are simple models which can be simulated, and experiments can probe them in exquisite detail since they couple to a magnetic field. Monte Carlo simulations are very useful for spin glasses. In this talk I showed: Finite temperature transition in 3-d Ising SG is well understood. There is a finite-temperature transition in the three-dimensional Heisenberg spin glass. The spin-glass and chiral-glass transition temperatures are very close and may well be equal. There does not appear to be an Almeida-Thouless line in three dimensions, though it may occur for d greater than a critical value (perhaps 6). (The last two are not yet universally accepted.) Thank You p.27