Phase Transitions in Spin Glasses

Transcription

1 p.1 Phase Transitions in Spin Glasses Peter Young peter/talks/bifi2008.pdf Work supported by the and the Hierarchical Systems Research Foundation. Collaborators: H. G. Katzgraber, L. W. Lee, and J. Pixley Talk at BIFI 2008, February 6 8, 2008, Zaragoza

2 p.2 Overview Basic Introduction What is a spin glass? experiments, theory Numerical techniques and finite size scaling Try to answer a long-standing question for spin glasses Is there a phase transition in an isotropic 2-component (XY) spin glass?

3 What is a spin glass? A system with disorder and frustration. Most theory uses the simplest model with these ingredients: the Edwards-Anderson Model: or H = i,j J ij S i S j. [J ij ] av = 0; [J 2 ij ]1/2 av = J (= 1) The S i have m components (call m 2 a vector SG): m = 1 (Ising) m = m = 2 (XY) 3 (Heisenberg). Will take a Gaussian distribution for the J ij. p.3

4 Compare with simpler systems Ferromagnet: All J ij > 0. Ground state: All spins parallel. i.e. Ground state is trivial. Antiferromagnet: All (nearest neighbor) J ij < 0 on a bipartite lattice. Ground state: Neighboring spins anti-parallel. i.e. Ground state is trivial. Spin Glass: Random mixture of ferro and anti-ferro. Because of frustration and disorder even the determining the ground state is a non-trivial optimization problem. p.4

5 p.5 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. Insulators: e.g. Fe 0.5 Mn 0.5 TiO 3. Hexagonal layers. Spins align perpendicular to layers (hence Ising-like). Within a layer: FeTiO 3 spins ferromagnetically coupled MnTiO 3 spins antiferromagnetically coupled. Hence mixture gives an Ising spin glass with short range interactions.

6 Related Problems Connection to other problems: Protein folding. Analogies with structural glasses (e.g. window glass)? Error correcting codes. Optimization problems in computer science. Exchange of ideas has gone both ways: Sophisticated algorithms from computer science have been used to find exact ground states of spins glasses for large sizes (mainly in two-dimensions). Ideas from spin glasses have proved very fruitful in solving satisfiability problems of interest to computer scientists. Spin glasses are a particularly convenient system with which to study glassy behavior because: 1. There are experimental systems which can be probed in detail with magnetic field. 2. There are simple theoretical models which can be easily simulated. p.6

7 p.7 Spin Glass Phase Transition Phase transition at T = T SG. For T < T SG the spin freeze in some random-looking orientation. The order parameter is the overlap between spin configurations of two identical copies of the system: q = 1 N N i=1 S (1) i S (2) i.

8 p.8 Spin Glass Phase Transition II Phase transition at T = T SG. 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.

9 p.9 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%. For Mn, the anisotropy is small (but non-zero). Is this anisotropy essential to get the observed transition?

10 p.10 Slow Dynamics Slow dynamics The dynamics is very slow at System low not T. in equilibrium due to complicated energy landscape: system trapped in one valley for long times. (free) energy valley barrier E configuration valley

11 p.11 Aging χ (ω) and aging The main figure shows data for χ (ω) for a spin glass below T SG by Jönsson et al. (2002). The sample is quenched, i.e. cooled rapidly, to below T SG and the ac susceptibility at ω/2π = 0.17,0.51, 1.7, 5.1, 17,55, 170 χ (ω) (arb. units) T (K) t (s) x 10 4 Hz (from bottom to top), is measured as a function of T = 17 K 500 time t (s) ω/2π = 0.17 Hz Note the frequency dependence and the aging with time. Aging is clear evidence that the system is not in equilibrium.

12 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.12

13 Theory 1 Mean Field Theory (Edwards-Anderson, Sherrington-Kirkpatrick, Parisi). Exact solution of an infinite range (SK) model. It has a finite T SG. The Parisi solution is a tour-de-force. Has an infinite number of order parameters. More than a quarter of a century after it was obtained using the replica trick, (needed replica symmetry breaking (RSB)) Tallegrand proved rigorously! that the Parisi free energy is exact for the SK model. H AT line For Ising SG, there is a transition (Almeida-Thouless (AT) line) in a field. 0 RSB solution (Parisi) (complicated) RS solution (simple) T T c p.13

14 p.14 Theory 2 Short-range (EA) models. Simulations on Ising systems also indicate a finite T SG (see later) in d = 3. Vector spin glasses? (See later.)

15 p.14 Theory 2 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:

16 p.14 Theory 2 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 (RSB), (Parisi). Droplet picture (DP) (Fisher and Huse, also Bray and Moore, and McMillan). Assume short-range is similar to infinite-range. There is a line of transitions in a magnetic field called the AT line. Focus on the geometrical aspects of the low-energy excitations. No AT line.

17 p.15 Problems of Interest The following problems of interest have been tackled by numerical simulations: 1. Is there a phase transition in zero field? 2. In a finite field, is there an AT line? 3. What is the nature of the spin glass phase (RSB, or droplet or something else)? 4. Can we understand the observed non-equilibrium phenomena? Here, I will just discuss question 1.

18 p.16 Overview Basic Introduction What is a spin glass? experiments, theory Numerical techniques and finite size scaling Try to answer a long-standing question for spin glasses Is there a phase transition in an isotropic 2-component (XY) spin glass?

19 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. Simple to implement. Acceptance probability satisfies the detailed balance condition T Result: temperature of each copy performs a random walk between T 1 and T n. p.17

20 p.18 Parallelization Embarassingly parallel: run each sample on a different processor. No interprocessor communication needed at all. However, such runs can take a very long time. Hence we sometimes use one processor for one temperature for one sample (so one sample is spread over many processors). Interprocessor communication is now present but small.

21 p.19 Correlation Length Spin glass correlation Function χ SG (k) = 1 N [ S i S j 2 ] av e ik (R i R j ) i,j Note: χ nl χ SG (k = 0), which is essentially the correlation volume. Determine the finite-size spin glass correlation length ξ L from the Ornstein Zernicke equation: χ SG (0) χ SG (k) = 1 + ξl 2k2 +..., by fitting to k = 0 and k = k min = 2π L (1, 0, 0). First applied to spin glasses by Ballesteros et al. (2000).

22 p.20 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...

23 p.21 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

24 p.22 Overview Basic Introduction What is a spin glass? experiments, theory Numerical techniques and finite size scaling Try to answer a long-standing question for spin glasses Is there a phase transition in an 2-component (XY) spin glass?

25 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) κ µ i = 1 2 sgn(j lm ) sin(θ l θ m ), XY (µ square) 2 l,m XY p.23

26 p.24 Motivation for vector model Most theory done for the Ising (S i = ±1) spin glass. Clear evidence for finite T SG. Best evidence: finite size scaling (FSS) of correlation length (Ballesteros et al.)

27 p.24 Motivation for vector model Most theory done for the Ising (S i = ±1) spin glass. Clear evidence for finite T SG. Best evidence: finite size scaling (FSS) of correlation length (Ballesteros et al.) Many experiments closer to a vector spin glass S i. Theoretical situation less clear:

28 p.24 Motivation for vector model Most theory done for the Ising (S i = ±1) spin glass. Clear evidence for finite T SG. Best evidence: finite size scaling (FSS) of correlation length (Ballesteros et al.) Many experiments closer to a vector spin glass S i. Theoretical situation less clear: Old Monte Carlo: T SG, if any, seems very low, probably zero.

29 p.24 Motivation for vector model Most theory done for the Ising (S i = ±1) spin glass. Clear evidence for finite T SG. Best evidence: finite size scaling (FSS) of correlation length (Ballesteros et al.) Many experiments closer to a vector spin glass S i. Theoretical situation less clear: Old Monte Carlo: 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.

30 p.24 Motivation for vector model Most theory done for the Ising (S i = ±1) spin glass. Clear evidence for finite T SG. Best evidence: finite size scaling (FSS) of correlation length (Ballesteros et al.) Many experiments closer to a vector spin glass S i. Theoretical situation less clear: Old Monte Carlo: 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. But: possibility of finite T SG raised by various authors, e.g. Maucourt &Grempel, Akino & Kosterlitz, Granato, Matsubara et al. Nakamura and Endoh (non-equilibrium MC) proposed a single transition for spins and chiralities.

31 p.24 Motivation for vector model Most theory done for the Ising (S i = ±1) spin glass. Clear evidence for finite T SG. Best evidence: finite size scaling (FSS) of correlation length (Ballesteros et al.) Many experiments closer to a vector spin glass S i. Theoretical situation less clear: Old Monte Carlo: 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. But: possibility of finite T SG raised by various authors, e.g. Maucourt &Grempel, Akino & Kosterlitz, Granato, Matsubara et al. Nakamura and Endoh (non-equilibrium MC) proposed a single transition for spins and chiralities. Hence confusing.

32 p.24 Motivation for vector model Most theory done for the Ising (S i = ±1) spin glass. Clear evidence for finite T SG. Best evidence: finite size scaling (FSS) of correlation length (Ballesteros et al.) Many experiments closer to a vector spin glass S i. Theoretical situation less clear: Old Monte Carlo: 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. But: possibility of finite T SG raised by various authors, e.g. Maucourt &Grempel, Akino & Kosterlitz, Granato, Matsubara et al. Nakamura and Endoh (non-equilibrium MC) proposed a single transition for spins and chiralities. Hence confusing. This work: look at FSS of the correlation lengths of both spins and chiralities for the XY spin glass. Useful because

33 p.24 Motivation for vector model Most theory done for the Ising (S i = ±1) spin glass. Clear evidence for finite T SG. Best evidence: finite size scaling (FSS) of correlation length (Ballesteros et al.) Many experiments closer to a vector spin glass S i. Theoretical situation less clear: Old Monte Carlo: 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. But: possibility of finite T SG raised by various authors, e.g. Maucourt &Grempel, Akino & Kosterlitz, Granato, Matsubara et al. Nakamura and Endoh (non-equilibrium MC) proposed a single transition for spins and chiralities. Hence confusing. This work: look at FSS of the correlation lengths of both spins and chiralities for the XY spin glass. Useful because this was the most successful approach for the Ising spin glass.

34 Motivation for vector model Most theory done for the Ising (S i = ±1) spin glass. Clear evidence for finite T SG. Best evidence: finite size scaling (FSS) of correlation length (Ballesteros et al.) Many experiments closer to a vector spin glass S i. Theoretical situation less clear: Old Monte Carlo: 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. But: possibility of finite T SG raised by various authors, e.g. Maucourt &Grempel, Akino & Kosterlitz, Granato, Matsubara et al. Nakamura and Endoh (non-equilibrium MC) proposed a single transition for spins and chiralities. Hence confusing. This work: look at FSS of the correlation lengths of both spins and chiralities for the XY spin glass. Useful because this was the most successful approach for the Ising spin glass. treat spins and chiralities on equal footing. p.24

35 p.25 XY Spin Glass (J. Pixley and A. P. Young (unpublished)). For smaller sizes, the data intersect at a common temperature and splay out at lower T (agrees with Lee and APY (2003)). However, for larger sizes the results at low-t seem rather marginal. (Similar effect found earlier for Heisenberg case, Campos et al. (2006), Lee and APY (2007)). Note, though, that, the effect is almost identical for both the spin-glass and the chiral-glass correlation lengths. No evidence for spin-chirality decoupling.

36 p.26 Conclusions Understanding phase transitions in spin glasses is difficult. Because of slow dynamics the range of sizes studied is rather limited, even using parallel tempering. Finite size scaling is essential to try to extrapolate to the thermodynamic limit, but there are large corrections to scaling for vector spin glasses. Why? Evidence for a sharp phase transition seems unambiguous for the Ising case. For vector spin glasses, there appears to be a transition of a marginal type (d=3 is close to the lower critical dimension ).

37 p.26 Conclusions Understanding phase transitions in spin glasses is difficult. Because of slow dynamics the range of sizes studied is rather limited, even using parallel tempering. Finite size scaling is essential to try to extrapolate to the thermodynamic limit, but there are large corrections to scaling for vector spin glasses. Why? Evidence for a sharp phase transition seems unambiguous for the Ising case. For vector spin glasses, there appears to be a transition of a marginal type (d=3 is close to the lower critical dimension ). Need even larger sizes for vector spin glasses to clarify the situation