New insights from one-dimensional spin glasses.

Transcription

1 New insights from one-dimensional spin glasses 1

2 New insights from one-dimensional spin glasses Helmut G. Katzgraber 2

3 New insights from one-dimensional spin glasses Helmut G. Katzgraber 2

4 Outline and Motivation Introduction to spin glasses (disordered magnets) What are spin glasses? Why are they hard to study? + + _ How well does the mean-field solution work? Model: 1D chain Do spin glasses order in a field? Ultrametricity in spin glasses? Applications to other problems and algorithm benchmarking. Work done in collaboration with W. Barthel, S. Böttcher, B. Gonçalves, A. K. Hartmann, T. Jörg, F. Krzakala, and A. P. Young. H AT H SG PM Tc T + 3

5 Outline and Motivation Introduction to spin glasses (disordered magnets) What are spin glasses? Why are they hard to study? + + _ How well does the mean-field solution work? Model: 1D chain Do spin glasses order in a field? Ultrametricity in spin glasses? Applications to other problems and algorithm benchmarking. Work done in collaboration with W. Barthel, S. Böttcher, B. Gonçalves, A. K. Hartmann, T. Jörg, F. Krzakala, and A. P. Young. H AT H SG PM? Tc T + 3

6 Outline and Motivation Introduction to spin glasses (disordered magnets) What are spin glasses? Why are they hard to study? + + _ How well does the mean-field solution work? Model: 1D chain Do spin glasses order in a field? Ultrametricity in spin glasses? Applications to other problems and algorithm benchmarking. Work done in collaboration with W. Barthel, S. Böttcher, B. Gonçalves, A. K. Hartmann, T. Jörg, F. Krzakala, and A. P. Young. H AT H SG PM? Tc T + 3

7 Brief introduction to spin glasses 4

8 Building a spin glass from the Ising model Hamiltonian: H = ij J ij S i S j H i S i m J ij =1 i, j i j Order parameter: m = 1 S i (magnetization) N i Some properties: Phase transition to an ordered state. According to Harris criterion, if dν > 2 the system changes universality class when local disorder is added. For the Ising model dν 2 in 2D and 3D. T c H T 5

9 What is a spin glass? 2x2 Prototype model: Edwards-Anderson Ising spin glass ferromagnet Properties: H = ij disorder frustration J ij S i S j h S i Phase transition into a glassy phase Complex energy landscape Some hallmarks: aging, memory, hysteresis,... Complex optimization problem + Very hard to treat analytically beyond mean-field. +? + spin glass! χ ac J ij random E χ ac configuration space ω 1 ω 2 ω 3 t T w T c T M H 6

10 What is a spin glass? 2x2 Prototype model: Edwards-Anderson Ising spin glass ferromagnet Properties: H = ij disorder frustration J ij S i S j h S i Phase transition into a glassy phase Complex energy landscape Some hallmarks: aging, memory, hysteresis,... Complex optimization problem + Very hard to treat analytically beyond mean-field. +? + spin glass! χ ac J ij random E χ ac configuration space ω 1 ω 2 ω 3 t T w T c T M H 6

11 What is a spin glass? 2x2 Prototype model: Edwards-Anderson Ising spin glass ferromagnet Properties: H = ij disorder frustration J ij S i S j h S i Phase transition into a glassy phase Complex energy landscape Some hallmarks: aging, memory, hysteresis,... Complex optimization problem + Very hard to treat analytically beyond mean-field. +? + spin glass! χ ac J ij random E χ ac configuration space ω 1 ω 2 ω 3 t T w T c T M Many H applications to other problems! 6

12 Applications beyond disordered magnets The models can describe many systems with competing interactions on a graph: Computer chips: S i J ij Economic markets: S i J ij Other applications: component wiring diagram agent inclination Quantum error correction in payload distribution topological quantum computing (current research). Optimization problems (e.g., number partitioning problem). Neural networks, portfolio interactions chip optimization markets 7

13 Experimental discovery, theoretical pictures Early experimental observations: 1970: Canella and Mydosh see a cusp in the susceptibility of Fe/Au alloys (disorder). Material with RKKY interactions (frustration): C J ij cos(2k FR ij )! R 3 ij Early theoretical descriptions: 1975: Introduction of the Edwards-Anderson Ising spin-glass model ( J ij random): H = J ij S i S j mean-field approx. ij 1975: Mean-field Sherrington-Kirkpatrick model. 1979: Parisi solution of the mean-field model (RSB). 1986: Fisher, Huse, Bray, Moore introduce the phenomenological droplet picture (DP) for short-range systems. T c T T c ij i,j T 8

14 How can we study these systems? Analytically: only the mean-field solution (RSB) or qualitative descriptions (DP). Numerically: Optimal problem for large computers Challenges: Exponential number of competing states (usually NP hard). Relaxation times diverge exponentially with the system size. Extra overhead due to disorder averaging. Usually small systems only. Solution: Use large computer clusters. Use better algorithms. Use better models. brutus Average project requires CPUh (4 months on 10 2 CPUs). 9

15 How can we study these systems? Analytically: only the mean-field solution (RSB) or qualitative descriptions (DP). Numerically: Optimal problem for large computers Challenges: Exponential number of competing states (usually NP hard). Relaxation times diverge exponentially with the system size. Extra overhead due to disorder averaging. Usually small systems only. Solution: Use large computer clusters. talk I Use better algorithms. Use better models. brutus Average project requires CPUh (4 months on 10 2 CPUs). 9

16 How can we study these systems? Analytically: only the mean-field solution (RSB) or qualitative descriptions (DP). Numerically: Optimal problem for large computers Challenges: Exponential number of competing states (usually NP hard). Relaxation times diverge exponentially with the system size. Extra overhead due to disorder averaging. Usually small systems only. Solution: Use large computer clusters. Use better algorithms. talk II Use better models. brutus Average project requires CPUh (4 months on 10 2 CPUs). 9

17 Some open problems... many challenges Some open issues: H 1 P(J) q H AT SG PM ultrametricity 0! AC J universality AT line Tc T And many more: Chaos in spin glasses Nature of the low-temperature spin-glass phase Properties of vector spin glasses, T w T c memory effect T 10

18 Some open problems... many challenges Some open issues: H 1 P(J) q H AT SG PM ultrametricity 0! AC J universality AT line Tc T And many more: Chaos in spin glasses Nature of the low-temperature spin-glass phase Properties of vector spin glasses, T w T c memory effect T 10

19 Some open problems... many challenges Some open issues: H 1 P(J) q H AT SG PM ultrametricity 0! AC J universality AT line Tc T And many more: Chaos in spin glasses Nature of the low-temperature spin-glass phase Properties of vector spin glasses, Which properties of the mean-field solution carry over to short-range systems? T w T c memory effect T 10

20 Models: The 1D Ising chain 11

21 Traditional model: Edwards-Anderson Hamiltonian: P(J ij ) H = i,j J ij S i S j S i {±1} Details about the model: Nearest-neighbor interactions Simulations usually done with periodic boundary conditions. Transition temperatures: Tc = 0 (2D), Tc ~ 1 (3D), Tc ~ 2 (4D). Most studied spin-glass model to date. J ij Disadvantages of the model: Cannot be solved analytically. In high space dimensions only small systems can be simulated (D 5 almost impossible). 12

22 Better: The one-dimensional Ising chain h i S i d d = 2σ + IR MF LR σ c (d) SR H = ij Properties: J ij S i S j i The sum ranges over all spins Power-law random interactions d l SK T c > 0 T c > 0 J ij ε ij r σ ij r ij 1 LR 0 d = σ SR Advantages: Large range of sizes. small large T c = 0 T c = 0 1/2 1 2 σ σ Fisher & Huse, PRB (88) d eff Kotliar et al., PRB (83) Tuning the power-law exponent changes the universality class. 13

23 Better: The one-dimensional Ising chain h i S i d d = 2σ + IR MF LR σ c (d) SR H = ij Properties: J ij S i S j i The sum ranges over all spins Power-law random interactions d l SK T c > 0 T c > 0 J ij ε ij r σ ij r ij 1 LR 0 d = σ SR Advantages: Large range of sizes. large small T c = 0 T c = 0 1/2 1 2 σ σ Fisher & Huse, PRB (88) d eff Kotliar et al., PRB (83) Tuning the power-law exponent changes the universality class. 13

24 Better: The one-dimensional Ising chain h i S i d d = 2σ + IR MF LR σ c (d) SR H = ij Properties: J ij S i S j i The sum ranges over all spins Power-law random interactions d l SK T c > 0 T c > 0 J ij ε ij r σ ij r ij 1 LR 0 d = σ SR Advantages: Large range of sizes. large small T c = 0 T c = 0 1/2 1 2 σ σ Fisher & Huse, PRB (88) d eff Kotliar et al., PRB (83) Tuning the power-law exponent changes the universality class. 13

25 Tuning the universality class Short-range spin glasses: Upper critical dimension du = 6 (for d du MF behavior) Lower critical dimension dl = 2 (for d dl Tc = 0) d eff 2 2σ 1 Phase diagram of the 1D chain: Kotliar et al., PRB (83) σ = 0 1/2 2/3 1 SK MF non-mf d eff = T c > 0 T c =

26 Tuning the universality class Short-range spin glasses: Upper critical dimension du = 6 (for d du MF behavior) Lower critical dimension dl = 2 (for d dl Tc = 0) d eff 2 2σ 1 Phase diagram of the 1D chain: Kotliar et al., PRB (83) σ = 0 1/2 2/3 1 SK MF non-mf d eff = T c > 0 T c =0 6 2 expect MF predictions to work 14

27 Algorithms 15

28 Reminder: Exchange Monte Carlo Efficient algorithm to treat spin glasses at finite T. Idea: Simulate M copies of the system at different temperatures with Tmax > Tc (typically Tmax ~2Tc MF ). Allow swapping of neighboring temperatures: easy crossing of barriers. Hukushima & Nemoto (96) fast T M E P(E) configuration space T T 1 2 E slow T T 2 1 Extremely fast equilibration at low temperatures. For the following applications we use this algorithm... 16

29 Do spin glasses order in a field? 17

30 Spin-glass state in a field? Two possible scenarios: Replica Symmetry Breaking (RSB): existence of an instability line [Almeida & Thouless (78)] for the mean-field SK model. Droplet picture (DP): there is no spin-glass state in a field. H H RSB DP H AT PM PM SG SG Why should we care? T c T T c T The question lies at the core of theoretical descriptions. Field terms are ubiquitous in applications/experiments. Experimentally, numerically, and theoretically controversial. 18

31 Spin-glass state in a field? Two possible scenarios: Replica Symmetry Breaking (RSB): existence of an instability line [Almeida & Thouless (78)] for the mean-field SK model. Droplet picture (DP): there is no spin-glass state in a field. H H RSB DP H AT PM? PM SG SG Why should we care? T c T T c T The question lies at the core of theoretical descriptions. Field terms are ubiquitous in applications/experiments. Experimentally, numerically, and theoretically controversial. 18

32 Spin-glass state in a field? Two possible scenarios: Outline: 1. Tool to probe transition 2. Review of old 3D results 3. Improved results in 1D Replica Symmetry Breaking (RSB): existence of an instability line [Almeida & Thouless (78)] for the mean-field SK model. Droplet picture (DP): there is no spin-glass state in a field. H H RSB DP H AT PM? PM SG SG Why should we care? T c T T c T The question lies at the core of theoretical descriptions. Field terms are ubiquitous in applications/experiments. Experimentally, numerically, and theoretically controversial. 18

33 Probing criticality: correlation length Use the finite-size correlation length to probe criticality in spin-glass systems: Wave-vector-dependent connected spin-glass susceptibility: χ SG (k) = 1 [ ( S i S j T S i T S j T ) 2] N dis eik(r i R j ) ij Perform an Ornstein-Zernicke approximation: [χ SG (k)/χ SG (0)] 1 = 1 + ξ 2 Lk 2 + O[(ξ L k) 4 ] Ballesteros et al. PRB (00) Compensate for finite-size effects and periodic boundaries: [ ] 1/2 1 χsg (0) ξ L = 2 sin(k min /2) χ SG (k min ) 1 Finite-size scaling: Better than Binder ratio. ξ L L = X ( ) L 1/ν [T T c ] 1. 19

34 Probing criticality: correlation length Use the finite-size correlation length to probe criticality in spin-glass systems: Wave-vector-dependent connected spin-glass susceptibility: χ SG (k) = 1 [ ( S i S j T S i T S j T ) 2] N dis eik(r i R j ) ij Perform an Ornstein-Zernicke approximation: [χ SG (k)/χ SG (0)] 1 = 1 + ξ 2 Lk 2 + O[(ξ L k) 4 ] Ballesteros et al. PRB (00) Compensate for finite-size effects and periodic boundaries: [ ] 1/2 1 χsg (0) ξ L = 2 sin(k min /2) χ SG (k min ) 1 Finite-size scaling: Better than Binder ratio. ξ L L = X ( ) L 1/ν [T T c ] 1. 19

35 Probing criticality: correlation length Use the finite-size correlation length to probe criticality in spin-glass systems: Wave-vector-dependent connected spin-glass susceptibility: χ SG (k) = 1 [ ( S i S j T S i T S j T ) 2] N dis eik(r i R j ) ij Perform an Ornstein-Zernicke approximation: [χ SG (k)/χ SG (0)] 1 = 1 + ξ 2 Lk 2 + O[(ξ L k) 4 ] Ballesteros et al. PRB (00) Compensate for finite-size effects and periodic boundaries: [ ] 1/2 1 χsg (0) ξ L = 2 sin(k min /2) χ SG (k min ) 1 Finite-size scaling: Better than Binder ratio. ξ L L = X ( ) L 1/ν [T T c ] 1. 19

36 How well does this work for zero field? Study the 3D model: Katzgraber et al., PRB (06) H = i,j J ij S i S j Remember: ξ L L = X[L 1/ν (T T c )] The data cross at T c Spin-glass state at zero field. Next: apply a field 2. 20

37 How well does this work for zero field? Study the 3D model: Katzgraber et al., PRB (06) H = i,j J ij S i S j Remember: ξ L L = X[L 1/ν (T T c )] The data cross at T c Spin-glass state at zero field. Next: apply a field 2. 20

38 Old 3D results in a field Perform slices at different horizontal fields. H Katzgraber & Young, PRL (04) H = 0.3 H AT H H i SG PM h i S i Solution: Use 1D chain Tc T Can probe large systems. Can probe MF region to check if the method works

39 Old 3D results in a field Perform slices at different horizontal fields. H Katzgraber & Young, PRL (04) H = 0.3 H AT H H i SG PM h i S i Solution: Use 1D chain Tc T Can probe large systems. Can probe MF region to check if the method works

40 Old 3D results in a field Perform slices at different horizontal fields. H Katzgraber & Young, PRL (04) H = 0.3 H AT H H i SG PM h i S i Solution: Use 1D chain Tc T Can probe large systems. Can probe MF region to check if the method works

41 Old 3D results in a field Perform slices at different horizontal fields. H Katzgraber & Young, PRL (04) H = 0.3 H AT H H i SG PM h i S i Solution: Use 1D chain Tc T Can probe large systems. Can probe MF region to check if the method works

42 Old 3D results in a field Perform slices at different horizontal fields. H Katzgraber & Young, PRL (04) H = 0.3 H = 0.05 H AT H H i SG PM h i S i Solution: Use 1D chain Tc T Can probe large systems. Can probe MF region to check if the method works

43 Old 3D results in a field Perform slices at different horizontal fields. Problem: H H H small systems. h i S i i H AT SG PM Solution: Use 1D chain Maybe AT line for d du? Tc T Can probe large systems. Can probe MF region to check if the method works.?? No AT line in 3D.? Katzgraber & Young, PRL (04) H = 0.3 H = 0.05 Does the method pick up the AT line?? 2. 21

44 Old 3D results in a field Perform slices at different horizontal fields. Problem: H H H small systems. h i S i i H AT SG PM Solution: Use 1D chain Maybe AT line for d du? Tc T Can probe large systems. Can probe MF region to check if the method works.?? No AT line in 3D.? Katzgraber & Young, PRL (04) H = 0.3 H = 0.05 Does the method pick up the AT line?? 2. 21

45 Tuning the universality class (1D chain) External field H = The data span a large range of system sizes. The AT line vanishes outside the MF regime ( σ 2/3). Katzgraber & Young, PRB (2005) 0 1/2 2/3 1 SK MF non-mf σ AT line AT 6 2 no AT d 3. 22

46 Tuning the universality class (1D chain) External field H = The data span a large range of system sizes. The AT line vanishes outside the MF regime ( σ 2/3). Katzgraber & Young, PRB (2005) 0 1/2 2/3 1 SK MF non-mf σ AT 6 2 no AT d 3. 22

47 Tuning the universality class (1D chain) External field H = The data span a large range of system sizes. The AT line vanishes outside the MF regime ( σ 2/3). Katzgraber & Young, PRB (2005) 0 1/2 2/3 1 SK MF non-mf σ no AT line AT 6 2 no AT d 3. 22

48 Tuning the universality class (1D chain) External field H = The data span a large range of system sizes. The AT line vanishes outside the MF regime ( σ 2/3). Katzgraber & Young, PRB (2005) 0 1/2 2/3 1 SK MF non-mf σ no AT line AT 6 2 no AT d 3. 22

49 Spin-glass state in a field? The AT line vanishes when not in the mean-field regime. For short-range spin glasses below the upper critical dimension: H H H H AT H AT PM PM PM SG SG SG Tc T Tc T T c T Does the behavior change for even larger system sizes? What happens for narrow AT lines? (see cond-mat/ ) 23

50 Spin-glass state in a field? The AT line vanishes when not in the mean-field regime. For short-range spin glasses below the upper critical dimension: H H H AT PM PM SG SG Tc T Tc T Does the behavior change for even larger system sizes? What happens for narrow AT lines? (see cond-mat/ ) 23

51 Spin-glass state in a field? The AT line vanishes when not in the mean-field regime. For short-range spin glasses below the upper critical dimension: H H H AT PM PM SG SG Tc T Tc T Does the behavior change for even larger system sizes? What happens for narrow AT lines? (see cond-mat/ ) 23

52 Ultrametricity in spin glasses 24

53 What is ultrametricity? Formal definition: Given a metric space with a distance function. In general, the distance function obeys the triangle inequality: d(α, γ) d(α, β)+d(β, γ) β In an ultrametric space we have a stronger inequality: d(α, γ) max{d(α, β),d(β, γ)} Note: Every triangle is isosceles in an ultrametric space. Examples: Linguistics (space where words differ) Taxonomy (classification of species). Number theory (p-adic numbers),... α γ 25

54 What is ultrametricity? Formal definition: Given a metric space with a distance function. In general, the distance function obeys the triangle inequality: d(α, γ) d(α, β)+d(β, γ) β In an ultrametric space we have a stronger inequality: d(α, γ) max{d(α, β),d(β, γ)} Outline: 1. Definitions 2. Tools 3. Results Note: Every triangle is isosceles in an ultrametric space. Examples: Linguistics (space where words differ) Taxonomy (classification of species). Number theory (p-adic numbers),... α γ 25

55 Relevance to spin glasses (hand-waving ) Replica symmetry breaking solution of the mean-field ( model: Z n ) 1 F = kt[log Z] av log Z = lim n 0 n Order parameter: overlap function q = 1 N Si α S β i q = 1 α N N i=1 After replication one obtains: [log Z] av Π α,β dq αβ e NG(Q αβ) Typical structure: Q αβ = 1 N S (α) i S (β) i = N i=1 One can show that for three states in Q αβ with q αγ q γβ q αβ one has q γβ = q αβ in the thermodynamic limit. β 1 α, β n Parisi (79) Talagrand (06) 1. 26

56 What does it mean to be ultrametric? Simple test to see if the mean-field solution is applicable to a model: Ultrametricity is a cornerstone of the mean-field solution. If a model has no ultrametricity, the RSB solution is not valid for it. Current state of affairs: Are short-range models ultrametric? Very controversial! Many contradicting predictions. Problems: Only small short-range systems can be studied. The states for the test have to be selected very carefully. Hed et al., PRL (04) Contucci et al., PRL (06) Jörg & Krzakala, PRL (07) and many more... β Solution: Analysis of the 1D chain [similar to Hed et al. (04)]. α γ 1. 27

57 Achtung! Problems when picking 3 states Possible pitfalls: 1. If time-reversal symmetry is unbroken, one has to ensure that all three states used belong to the same side of phase space. 2. The temperature must be much smaller than Tc. Hed et al., PRL (04) 3. If the temperature is too small, for large systems most triangles are equilateral (carry no information). Do not study too low T s. Solutions: 1. Can be avoided with a clustering analysis: Pick 3 states only from the left tree. 2. Simulations done at T < Tc, but not below T = 0.2Tc. Data shown for T = 0.4Tc. 3. To avoid equilateral triangles pick the 3 states from different branches in the left subtree (C1a, C1b, C2). Next 2. 28

58 Selection of states (similar to Hed et al.) Generation of states: Equilibrate system. Store 10 3 states per realization. Selection of states: Sorted dendrograms using Wards clustering method. Pick left tree ( spin down ). Split left tree into C1 C2. Split C1 into C1a and C1b. Pick three random states: α C 1a, β C 1b, γ C 2. Distance matrix: Darker colors mean closer distances d αβ = (1 q αβ )/2. s! states C 1a C 1 "C 2 C 1 dendrogram C 1b C states A T < Tc B distance matrix from Hed et al., PRL (04) 2. 29

59 Typical distance matrices at T < Tc Data for the 1D chain, L = 512, T = 0.4Tc. SK σ =0.75 Darker colors correspond to closer hamming distances. Both systems show structure at low temperatures

60 Typical distance matrices at T < Tc Data for the 1D chain, L = 512, T = 0.4Tc. SK σ =0.75 T > T c Darker colors correspond to closer hamming distances. Both systems show structure at low temperatures

61 Typical distance matrices at T < Tc Data for the 1D chain, L = 512, T = 0.4Tc. SK σ =0.75 Darker colors correspond to closer hamming distances. Both systems show structure at low temperatures

62 How to measure ultrametricity Observable: Select three states with the aforementioned recipe. Compute the hamming distance between them: d αβ = (1 q αβ )/2. Sort the distances: d max d med d min. Compute: K = d max d med ρ(d) Here ρ(d) Signs of ultrametricity: is the width of the distance distribution. If the space is ultrametric, we expect d max = d med for N. Study the distribution P(K). We expect: P (K) δ(k = 0) N 2. 31

63 How to measure ultrametricity Observable: Select three states with the aforementioned recipe. Compute the hamming distance between them: d αβ = (1 q αβ )/2. Sort the distances: d max d med d min. Compute: K = d max d med ρ(d) Here ρ(d) Signs of ultrametricity: is the width of the distance distribution. If the space is ultrametric, we expect d max = d med for N. Study the distribution P(K). We expect: P (K) δ(k = 0) N 2. 31

64 Distributions P(K) at T = 0.4Tc Katzgraber & Hartmann, submitted SK σ =0.75 Currently: try to determine the number of RSB layers and clusters. Similar results for other values of σ

65 Distributions P(K) at T = 0.4Tc Katzgraber & Hartmann, submitted SKSK σ =0.75 T = Tc Currently: try to determine the number of RSB layers and clusters. Similar results for other values of σ

66 Distributions P(K) at T = 0.4Tc Katzgraber & Hartmann, submitted SK σ =0.75 Currently: try to determine the number of RSB layers and clusters. Similar results for other values of σ

67 Distributions P(K) at T = 0.4Tc Katzgraber & Hartmann, submitted SK σ =0.75 indication of ultrametricity? Currently: try to determine the number of RSB layers and clusters. Similar results for other values of σ

68 33

69 What else can we do with the 1D chain? 34

70 Other ideas currently explored Study open problems in spin glasses: Chaos, aging, universality, memory effect,! AC Modifications of the model: Spin symmetries (Potts, Heisenberg, ). p-spin model for structural glasses. Matsuda et al. (07) Power-law probability-diluted chain (huge systems). T w T c T Benchmarking algorithms: How does the algorithm scale with the size of the input (N)? How does the scaling of the algorithm depend on the complexity/connectivity? 0 1/2 2/3 1 SK MF non-mf 6 2 1D chain: range of the interactions (universality class) can be changed. σ d 35

71 Other ideas currently explored Study open problems in spin glasses: Chaos, aging, universality, memory effect,! AC Examples: 1. diluted model 2. benchmarks Modifications of the model: Spin symmetries (Potts, Heisenberg, ). p-spin model for structural glasses. Matsuda et al. (07) Power-law probability-diluted chain (huge systems). T w T c T Benchmarking algorithms: How does the algorithm scale with the size of the input (N)? How does the scaling of the algorithm depend on the complexity/connectivity? 0 1/2 2/3 1 SK MF non-mf 6 2 1D chain: range of the interactions (universality class) can be changed. σ d 35

72 Probability-diluted Gaussian Ising chain Place a Gaussian random bond with Same behavior as the regular 1D chain, but [z] av =2ζ(2σ). z(σ =0.75) 5.22 P(J ij 0) r 2σ H = i<j J ij S i S j Leuzzi, et al. (08) σ =0.75 Note: Leuzzi et al. fix the connectivity (VB limit). Here SK limit

73 Probability-diluted Gaussian Ising chain Place a Gaussian random bond with Same behavior as the regular 1D chain, but [z] av =2ζ(2σ). z(σ =0.75) 5.22 P(J ij 0) r 2σ H = i<j J ij S i S j Leuzzi, et al. (08) σ =0.75 now revisiting the AT line Note: Leuzzi et al. fix the connectivity (VB limit). Here SK limit

74 Example: Hysteretic optimization Experiment: Do you have a CRT monitor or a non-lcd TV at home? Take a magnet and hold it to the screen. You are in trouble. Zarand et al., PRL (02) Solution: Call the technician. Make a degaussing coil and slowly do circles around the TV increasing the radius and distance. You have just hysteretically (and possibly also hysterically) demagnetized the TV screen. 37

75 Example: Hysteretic optimization Experiment: Do you have a CRT monitor or a non-lcd TV at home? Take a magnet and hold it to the screen. You are in trouble. Zarand et al., PRL (02) Solution: Call the technician. Make a degaussing coil and slowly do circles around the TV increasing the radius and distance. You have just hysteretically (and possibly also hysterically) demagnetized the TV screen. 37

76 Benchmarking: Hysteretic optimization E 0.5 m!0.5!1 H 1 Zarand et al., PRL (02) Idea: Minimize the energy of a system by successive demagnetization. Example: Ising spin glass H = J ij S i S j H ξ i S i ij i ξ i = ±1 random m = 1 ξ i S i N i m H sat γh sat adapted from Gonçalves & Boettcher (08) H E0 dh for details see Hartmann & Rieger book (04) 2. 38

77 Benchmarking: Hysteretic optimization E Zarand et al., PRL (02) Idea: Minimize the energy of a system by successive demagnetization. Example: Ising spin glass infinite range H = J ij S i S j H ξ i S i ij i ξ i = ±1 random m = 1 ξ i S i N i m H sat γh sat 0.5 m!0.5!1 H 1 adapted from Gonçalves & Boettcher (08) E0 H dh for details see Hartmann & Rieger book (04) The algorithm works best in the infiniterange regime

78 Benchmarking: Hysteretic optimization E Zarand et al., PRL (02) Idea: Minimize the energy of a system by successive demagnetization. Example: Ising spin glass infinite range H = J ij S i S j H ξ i S i ij i ξ i = ±1 random m = 1 ξ i S i N i m H sat γh sat 0.5 m!0.5!1 H 1 adapted from Gonçalves & Boettcher (08) E0 H dh for details see Hartmann & Rieger book (04) Other The algorithm works best in the infiniterange regime. algorithms tested: PT, hboa, BCP,

79 New insights from one-dimensional spin glasses 39

80 New insights from one-dimensional Summary: spin glasses Do short-range spin glasses order in a field? No. Are short-range spin glasses ultrametric? Seems like it. What does this mean? The mean-field solution works only for certain aspects. Advantages of the 1D model: Large systems can be studied. The effective space dimension can be changed. Benchmark model for optimization algorithms. 39

81 Bill Watterson New insights from one-dimensional Summary: spin glasses Do short-range spin glasses order in a field? No. Are short-range spin glasses ultrametric? Seems like it. What does this mean? The mean-field solution works only for certain aspects. Advantages of the 1D model: Large systems can be studied. The effective space dimension can be changed. Benchmark model for optimization algorithms. thanks! 39