New insights from one-dimensional spin glasses.
|
|
- Loren Willis
- 5 years ago
- Views:
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
Phase Transitions in Spin Glasses
Phase Transitions in Spin Glasses Peter Young Talk available at http://physics.ucsc.edu/ peter/talks/sinica.pdf e-mail:peter@physics.ucsc.edu Supported by the Hierarchical Systems Research Foundation.
More informationIs the Sherrington-Kirkpatrick Model relevant for real spin glasses?
Is the Sherrington-Kirkpatrick Model relevant for real spin glasses? A. P. Young Department of Physics, University of California, Santa Cruz, California 95064 E-mail: peter@physics.ucsc.edu Abstract. I
More informationPhase Transitions in Spin Glasses
p.1 Phase Transitions in Spin Glasses Peter Young http://physics.ucsc.edu/ peter/talks/bifi2008.pdf e-mail:peter@physics.ucsc.edu Work supported by the and the Hierarchical Systems Research Foundation.
More informationIs there a de Almeida-Thouless line in finite-dimensional spin glasses? (and why you should care)
Is there a de Almeida-Thouless line in finite-dimensional spin glasses? (and why you should care) Peter Young Talk at MPIPKS, September 12, 2013 Available on the web at http://physics.ucsc.edu/~peter/talks/mpipks.pdf
More informationLearning About Spin Glasses Enzo Marinari (Cagliari, Italy) I thank the organizers... I am thrilled since... (Realistic) Spin Glasses: a debated, inte
Learning About Spin Glasses Enzo Marinari (Cagliari, Italy) I thank the organizers... I am thrilled since... (Realistic) Spin Glasses: a debated, interesting issue. Mainly work with: Giorgio Parisi, Federico
More informationPhase Transition in Vector Spin Glasses. Abstract
Phase Transition in Vector Spin Glasses A. P. Young Department of Physics, University of California, Santa Cruz, California 95064 (Dated: June 3, 2004) Abstract We first give an experimental and theoretical
More informationSpin glasses and Adiabatic Quantum Computing
Spin glasses and Adiabatic Quantum Computing A.P. Young alk at the Workshop on heory and Practice of Adiabatic Quantum Computers and Quantum Simulation, ICP, rieste, August 22-26, 2016 Spin Glasses he
More informationPhase Transition in Vector Spin Glasses. Abstract
Phase Transition in Vector Spin Glasses A. P. Young Department of Physics, University of California, Santa Cruz, California 95064 (Dated: June 2, 2004) Abstract We first give an experimental and theoretical
More informationThe Last Survivor: a Spin Glass Phase in an External Magnetic Field.
The Last Survivor: a Spin Glass Phase in an External Magnetic Field. J. J. Ruiz-Lorenzo Dep. Física, Universidad de Extremadura Instituto de Biocomputación y Física de los Sistemas Complejos (UZ) http://www.eweb.unex.es/eweb/fisteor/juan
More informationEquilibrium Study of Spin-Glass Models: Monte Carlo Simulation and Multi-variate Analysis. Koji Hukushima
Equilibrium Study of Spin-Glass Models: Monte Carlo Simulation and Multi-variate Analysis Koji Hukushima Department of Basic Science, University of Tokyo mailto:hukusima@phys.c.u-tokyo.ac.jp July 11 15,
More informationThrough Kaleidoscope Eyes Spin Glasses Experimental Results and Theoretical Concepts
Through Kaleidoscope Eyes Spin Glasses Experimental Results and Theoretical Concepts Benjamin Hsu December 11, 2007 Abstract A spin glass describes a system of spins on a lattice (or a crystal) where the
More informationSpin glasses, where do we stand?
Spin glasses, where do we stand? Giorgio Parisi Many progresses have recently done in spin glasses: theory, experiments, simulations and theorems! In this talk I will present: A very brief introduction
More informationThe glass transition as a spin glass problem
The glass transition as a spin glass problem Mike Moore School of Physics and Astronomy, University of Manchester UBC 2007 Co-Authors: Joonhyun Yeo, Konkuk University Marco Tarzia, Saclay Mike Moore (Manchester)
More informationScaling Theory. Roger Herrigel Advisor: Helmut Katzgraber
Scaling Theory Roger Herrigel Advisor: Helmut Katzgraber 7.4.2007 Outline The scaling hypothesis Critical exponents The scaling hypothesis Derivation of the scaling relations Heuristic explanation Kadanoff
More informationThe Random Matching Problem
The Random Matching Problem Enrico Maria Malatesta Universitá di Milano October 21st, 2016 Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, 2016 1 / 15 Outline 1 Disordered Systems
More informationSolving the Schrödinger equation for the Sherrington Kirkpatrick model in a transverse field
J. Phys. A: Math. Gen. 30 (1997) L41 L47. Printed in the UK PII: S0305-4470(97)79383-1 LETTER TO THE EDITOR Solving the Schrödinger equation for the Sherrington Kirkpatrick model in a transverse field
More informationQuantum Annealing in spin glasses and quantum computing Anders W Sandvik, Boston University
PY502, Computational Physics, December 12, 2017 Quantum Annealing in spin glasses and quantum computing Anders W Sandvik, Boston University Advancing Research in Basic Science and Mathematics Example:
More informationLow-temperature behavior of the statistics of the overlap distribution in Ising spin-glass models
University of Massachusetts Amherst From the SelectedWorks of Jonathan Machta 2014 Low-temperature behavior of the statistics of the overlap distribution in Ising spin-glass models Matthew Wittmann B.
More informationPopulation Annealing Monte Carlo Studies of Ising Spin Glasses
University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations Dissertations and Theses 2015 Population Annealing Monte Carlo Studies of Ising Spin Glasses wenlong wang wenlong@physics.umass.edu
More informationOptimized statistical ensembles for slowly equilibrating classical and quantum systems
Optimized statistical ensembles for slowly equilibrating classical and quantum systems IPAM, January 2009 Simon Trebst Microsoft Station Q University of California, Santa Barbara Collaborators: David Huse,
More informationStudying Spin Glasses via Combinatorial Optimization
Studying Spin Glasses via Combinatorial Optimization Frauke Liers Emmy Noether Junior Research Group, Department of Computer Science, University of Cologne Dec 11 th, 2007 What we are interested in...
More informationThe ultrametric tree of states and computation of correlation functions in spin glasses. Andrea Lucarelli
Università degli studi di Roma La Sapienza Facoltà di Scienze Matematiche, Fisiche e Naturali Scuola di Dottorato Vito Volterra Prof. Giorgio Parisi The ultrametric tree of states and computation of correlation
More informationThe cavity method. Vingt ans après
The cavity method Vingt ans après Les Houches lectures 1982 Early days with Giorgio SK model E = J ij s i s j i
More informationarxiv: v1 [cond-mat.dis-nn] 25 Apr 2018
arxiv:1804.09453v1 [cond-mat.dis-nn] 25 Apr 2018 Critical properties of the antiferromagnetic Ising model on rewired square lattices Tasrief Surungan 1, Bansawang BJ 1 and Muhammad Yusuf 2 1 Department
More informationEnergy-Decreasing Dynamics in Mean-Field Spin Models
arxiv:cond-mat/0210545 v1 24 Oct 2002 Energy-Decreasing Dynamics in Mean-Field Spin Models L. Bussolari, P. Contucci, M. Degli Esposti, C. Giardinà Dipartimento di Matematica dell Università di Bologna,
More informationPhase Transitions (and their meaning) in Random Constraint Satisfaction Problems
International Workshop on Statistical-Mechanical Informatics 2007 Kyoto, September 17 Phase Transitions (and their meaning) in Random Constraint Satisfaction Problems Florent Krzakala In collaboration
More informationBranislav K. Nikolić
Interdisciplinary Topics in Complex Systems: Cellular Automata, Self-Organized Criticality, Neural Networks and Spin Glasses Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware,
More informationarxiv: v1 [cond-mat.dis-nn] 13 Jul 2015
Spin glass behavior of the antiferromagnetic Heisenberg model on scale free network arxiv:1507.03305v1 [cond-mat.dis-nn] 13 Jul 2015 Tasrief Surungan 1,3, Freddy P. Zen 2,3, and Anthony G. Williams 4 1
More informationarxiv:cond-mat/ v2 [cond-mat.dis-nn] 26 Sep 2006
Critical behavior of the three- and ten-state short-range Potts glass: A Monte Carlo study arxiv:cond-mat/0605010v2 [cond-mat.dis-nn] 26 Sep 2006 L. W. Lee, 1 H. G. Katzgraber, 2 and A. P. Young 1, 1 Department
More informationSurface effects in frustrated magnetic materials: phase transition and spin resistivity
Surface effects in frustrated magnetic materials: phase transition and spin resistivity H T Diep (lptm, ucp) in collaboration with Yann Magnin, V. T. Ngo, K. Akabli Plan: I. Introduction II. Surface spin-waves,
More informationMesoscopic conductance fluctuations and spin glasses
GdR Meso 2010 G Thibaut CAPRON Track of a potential energy landscape evolving on geological time scales B Mesoscopic conductance fluctuations and spin glasses www.neel.cnrs.fr Team project Quantum coherence
More informationarxiv:cond-mat/ v1 [cond-mat.dis-nn] 2 Sep 1998
Scheme of the replica symmetry breaking for short- ranged Ising spin glass within the Bethe- Peierls method. arxiv:cond-mat/9809030v1 [cond-mat.dis-nn] 2 Sep 1998 K. Walasek Institute of Physics, Pedagogical
More informationMetropolis Monte Carlo simulation of the Ising Model
Metropolis Monte Carlo simulation of the Ising Model Krishna Shrinivas (CH10B026) Swaroop Ramaswamy (CH10B068) May 10, 2013 Modelling and Simulation of Particulate Processes (CH5012) Introduction The Ising
More informationSolution-space structure of (some) optimization problems
Solution-space structure of (some) optimization problems Alexander K. Hartmann 1, Alexander Mann 2 and Wolfgang Radenbach 3 1 Institute for Physics, University of Oldenburg, 26111 Oldenburg, Germany 2
More informationarxiv: v1 [cond-mat.dis-nn] 23 Apr 2008
lack of monotonicity in spin glass correlation functions arxiv:0804.3691v1 [cond-mat.dis-nn] 23 Apr 2008 Pierluigi Contucci, Francesco Unguendoli, Cecilia Vernia, Dipartimento di Matematica, Università
More informationQuantum and classical annealing in spin glasses and quantum computing. Anders W Sandvik, Boston University
NATIONAL TAIWAN UNIVERSITY, COLLOQUIUM, MARCH 10, 2015 Quantum and classical annealing in spin glasses and quantum computing Anders W Sandvik, Boston University Cheng-Wei Liu (BU) Anatoli Polkovnikov (BU)
More informationPhase transition phenomena of statistical mechanical models of the integer factorization problem (submitted to JPSJ, now in review process)
Phase transition phenomena of statistical mechanical models of the integer factorization problem (submitted to JPSJ, now in review process) Chihiro Nakajima WPI-AIMR, Tohoku University Masayuki Ohzeki
More informationarxiv: v1 [cond-mat.dis-nn] 13 Oct 2015
Search for the Heisenberg spin glass on rewired square lattices with antiferromagnetic interaction arxiv:1510.03825v1 [cond-mat.dis-nn] 13 Oct 2015 asrief Surungan, Bansawang BJ, and Dahlang ahir Department
More informationNETADIS Kick off meeting. Torino,3-6 february Payal Tyagi
NETADIS Kick off meeting Torino,3-6 february 2013 Payal Tyagi Outline My background Project: 1) Theoretical part a) Haus master equation and Hamiltonian b) First steps: - Modelling systems with disordered
More informationPhase transitions and finite-size scaling
Phase transitions and finite-size scaling Critical slowing down and cluster methods. Theory of phase transitions/ RNG Finite-size scaling Detailed treatment: Lectures on Phase Transitions and the Renormalization
More informationarxiv:cond-mat/ v4 [cond-mat.dis-nn] 23 May 2001
Phase Diagram of the three-dimensional Gaussian andom Field Ising Model: A Monte Carlo enormalization Group Study arxiv:cond-mat/488v4 [cond-mat.dis-nn] 3 May M. Itakura JS Domestic esearch Fellow, Center
More informationJ ij S i S j B i S i (1)
LECTURE 18 The Ising Model (References: Kerson Huang, Statistical Mechanics, Wiley and Sons (1963) and Colin Thompson, Mathematical Statistical Mechanics, Princeton Univ. Press (1972)). One of the simplest
More informationLow T scaling behavior of 2D disordered and frustrated models
Low T scaling behavior of 2D disordered and frustrated models Enzo Marinari (Roma La Sapienza, Italy) A. Galluccio, J. Lukic, E.M., O. C. Martin and G. Rinaldi, Phys. Rev. Lett. 92 (2004) 117202. Additional
More informationPhase transitions in discrete structures
Phase transitions in discrete structures Amin Coja-Oghlan Goethe University Frankfurt Overview 1 The physics approach. [following Mézard, Montanari 09] Basics. Replica symmetry ( Belief Propagation ).
More informationPhase Transitions in Relaxor Ferroelectrics
Phase Transitions in Relaxor Ferroelectrics Matthew Delgado December 13, 2005 Abstract This paper covers the properties of relaxor ferroelectrics and considers the transition from the paraelectric state
More informationarxiv: v3 [cond-mat.dis-nn] 4 Jan 2018
The bimodal Ising spin glass in dimension two : the anomalous dimension η P. H. Lundow Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87, Sweden I. A. Campbell Laboratoire
More informationPhase Transitions and Critical Behavior:
II Phase Transitions and Critical Behavior: A. Phenomenology (ibid., Chapter 10) B. mean field theory (ibid., Chapter 11) C. Failure of MFT D. Phenomenology Again (ibid., Chapter 12) // Windsor Lectures
More informationFinding Ground States of SK Spin Glasses with hboa and GAs
Finding Ground States of Sherrington-Kirkpatrick Spin Glasses with hboa and GAs Martin Pelikan, Helmut G. Katzgraber, & Sigismund Kobe Missouri Estimation of Distribution Algorithms Laboratory (MEDAL)
More informationSlightly off-equilibrium dynamics
Slightly off-equilibrium dynamics Giorgio Parisi Many progresses have recently done in understanding system who are slightly off-equilibrium because their approach to equilibrium is quite slow. In this
More informationA variational approach to Ising spin glasses in finite dimensions
. Phys. A: Math. Gen. 31 1998) 4127 4140. Printed in the UK PII: S0305-447098)89176-2 A variational approach to Ising spin glasses in finite dimensions R Baviera, M Pasquini and M Serva Dipartimento di
More informationQuantum spin systems - models and computational methods
Summer School on Computational Statistical Physics August 4-11, 2010, NCCU, Taipei, Taiwan Quantum spin systems - models and computational methods Anders W. Sandvik, Boston University Lecture outline Introduction
More informationThermodynamic States in Finite Dimensional Spin Glasses
Thermodynamic States in Finite Dimensional Spin Glasses J. J. Ruiz-Lorenzo Dep. Física & ICCAEx (Univ. de Extremadura) & BIFI (Zaragoza) http://www.eweb.unex.es/eweb/fisteor/juan/juan_talks.html Instituto
More informationSpontaneous Symmetry Breaking
Spontaneous Symmetry Breaking Second order phase transitions are generally associated with spontaneous symmetry breaking associated with an appropriate order parameter. Identifying symmetry of the order
More informationRenormalization group and critical properties of Long Range models
Renormalization group and critical properties of Long Range models Scuola di dottorato in Scienze Fisiche Dottorato di Ricerca in Fisica XXV Ciclo Candidate Maria Chiara Angelini ID number 1046274 Thesis
More informationExtremal Optimization for Low-Energy Excitations of very large Spin-Glasses
Etremal Optimization for Low-Energy Ecitations of very large Spin-Glasses Stefan Boettcher Dept of Physics Collaborators: Senior: Allon Percus (Los Alamos) Paolo Sibani (Odense-DK) Michelangelo Grigni
More informationStatistical Physics of The Symmetric Group. Mobolaji Williams Harvard Physics Oral Qualifying Exam Dec. 12, 2016
Statistical Physics of The Symmetric Group Mobolaji Williams Harvard Physics Oral Qualifying Exam Dec. 12, 2016 1 Theoretical Physics of Living Systems Physics Particle Physics Condensed Matter Astrophysics
More informationGraphical Representations and Cluster Algorithms
Graphical Representations and Cluster Algorithms Jon Machta University of Massachusetts Amherst Newton Institute, March 27, 2008 Outline Introduction to graphical representations and cluster algorithms
More informationPHYSICAL REVIEW LETTERS
PHYSICAL REVIEW LETTERS VOLUME 76 4 MARCH 1996 NUMBER 10 Finite-Size Scaling and Universality above the Upper Critical Dimensionality Erik Luijten* and Henk W. J. Blöte Faculty of Applied Physics, Delft
More information3.320 Lecture 18 (4/12/05)
3.320 Lecture 18 (4/12/05) Monte Carlo Simulation II and free energies Figure by MIT OCW. General Statistical Mechanics References D. Chandler, Introduction to Modern Statistical Mechanics D.A. McQuarrie,
More informationRenormalization Group: non perturbative aspects and applications in statistical and solid state physics.
Renormalization Group: non perturbative aspects and applications in statistical and solid state physics. Bertrand Delamotte Saclay, march 3, 2009 Introduction Field theory: - infinitely many degrees of
More informationHigh-Temperature Criticality in Strongly Constrained Quantum Systems
High-Temperature Criticality in Strongly Constrained Quantum Systems Claudio Chamon Collaborators: Claudio Castelnovo - BU Christopher Mudry - PSI, Switzerland Pierre Pujol - ENS Lyon, France PRB 2006
More informationQuantum Monte Carlo Simulations in the Valence Bond Basis. Anders Sandvik, Boston University
Quantum Monte Carlo Simulations in the Valence Bond Basis Anders Sandvik, Boston University Outline The valence bond basis for S=1/2 spins Projector QMC in the valence bond basis Heisenberg model with
More informationPhysics on Random Graphs
Physics on Random Graphs Lenka Zdeborová (CNLS + T-4, LANL) in collaboration with Florent Krzakala (see MRSEC seminar), Marc Mezard, Marco Tarzia... Take Home Message Solving problems on random graphs
More informationPhase Transitions in the Coloring of Random Graphs
Phase Transitions in the Coloring of Random Graphs Lenka Zdeborová (LPTMS, Orsay) In collaboration with: F. Krząkała (ESPCI Paris) G. Semerjian (ENS Paris) A. Montanari (Standford) F. Ricci-Tersenghi (La
More informationDecimation Technique on Sierpinski Gasket in External Magnetic Field
Egypt.. Solids, Vol. (), No. (), (009 ) 5 Decimation Technique on Sierpinski Gasket in External Magnetic Field Khalid Bannora, G. Ismail and M. Abu Zeid ) Mathematics Department, Faculty of Science, Zagazig
More informationQuantum (spin) glasses. Leticia F. Cugliandolo LPTHE Jussieu & LPT-ENS Paris France
Quantum (spin) glasses Leticia F. Cugliandolo LPTHE Jussieu & LPT-ENS Paris France Giulio Biroli (Saclay) Daniel R. Grempel (Saclay) Gustavo Lozano (Buenos Aires) Homero Lozza (Buenos Aires) Constantino
More informationMulticanonical methods
Multicanonical methods Normal Monte Carlo algorithms sample configurations with the Boltzmann weight p exp( βe). Sometimes this is not desirable. Example: if a system has a first order phase transitions
More informationMean field theory for Heisenberg spin glasses
Mean field theory for Heisenberg spin glasses G. Toulouse, M. Gabay To cite this version: G. Toulouse, M. Gabay. Mean field theory for Heisenberg spin glasses. Journal de Physique Lettres, 1981, 42 (5),
More informationSpontaneous Spin Polarization in Quantum Wires
Spontaneous Spin Polarization in Quantum Wires Julia S. Meyer The Ohio State University with A.D. Klironomos K.A. Matveev 1 Why ask this question at all GaAs/AlGaAs heterostucture 2D electron gas Quantum
More informationPhysics 127b: Statistical Mechanics. Second Order Phase Transitions. The Ising Ferromagnet
Physics 127b: Statistical Mechanics Second Order Phase ransitions he Ising Ferromagnet Consider a simple d-dimensional lattice of N classical spins that can point up or down, s i =±1. We suppose there
More informationCritical properties of disordered XY model on sparse random graphs
Manuale di Identità Visiva Sapienza Università di Roma Sapienza Università di Roma Scuola di Dottorato Vito Volterra Dipartimento di Fisica Critical properties of disordered XY model on sparse random graphs
More informationA Tour of Spin Glasses and Their Geometry
A Tour of Spin Glasses and Their Geometry P. Le Doussal, D. Bernard, LPTENS A. Alan Middleton, Syracuse University Support from NSF, ANR IPAM: Random Shapes - Workshop I 26 March, 2007 Goals Real experiment
More informationSpin glasses, concepts and analysis of susceptibility. P.C.W. Holdsworth Ecole Normale Supérieure de Lyon
Spin glasses, concepts and analysis of susceptibility P.C.W. Holdsworth Ecole Normale Supérieure de Lyon 1. A (brief) review of spin glasses 2. Slow dynamics, aging phenomena 3. An illustration from a
More informationThe Replica Approach: Spin and Structural Glasses
The Replica Approach: Spin and Structural Glasses Daniel Sussman December 18, 2008 Abstract A spin glass is a type of magnetic ordering that can emerge, most transparently, when a system is both frustrated
More informationBose-Einstein condensation in Quantum Glasses
Bose-Einstein condensation in Quantum Glasses Giuseppe Carleo, Marco Tarzia, and Francesco Zamponi Phys. Rev. Lett. 103, 215302 (2009) Collaborators: Florent Krzakala, Laura Foini, Alberto Rosso, Guilhem
More informationMicrocanonical scaling in small systems arxiv:cond-mat/ v1 [cond-mat.stat-mech] 3 Jun 2004
Microcanonical scaling in small systems arxiv:cond-mat/0406080v1 [cond-mat.stat-mech] 3 Jun 2004 M. Pleimling a,1, H. Behringer a and A. Hüller a a Institut für Theoretische Physik 1, Universität Erlangen-Nürnberg,
More informationPhase Transitions in Disordered Systems: The Example of the 4D Random Field Ising Model
Phase Transitions in Disordered Systems: The Example of the 4D Random Field Ising Model Nikos Fytas Applied Mathematics Research Centre, Coventry University, United Kingdom Work in collaboration with V.
More informationPhysics 115/242 Monte Carlo simulations in Statistical Physics
Physics 115/242 Monte Carlo simulations in Statistical Physics Peter Young (Dated: May 12, 2007) For additional information on the statistical Physics part of this handout, the first two sections, I strongly
More informationNon-magnetic states. The Néel states are product states; φ N a. , E ij = 3J ij /4 2 The Néel states have higher energy (expectations; not eigenstates)
Non-magnetic states Two spins, i and j, in isolation, H ij = J ijsi S j = J ij [Si z Sj z + 1 2 (S+ i S j + S i S+ j )] For Jij>0 the ground state is the singlet; φ s ij = i j i j, E ij = 3J ij /4 2 The
More informationValence Bonds in Random Quantum Magnets
Valence Bonds in Random Quantum Magnets theory and application to YbMgGaO 4 Yukawa Institute, Kyoto, November 2017 Itamar Kimchi I.K., Adam Nahum, T. Senthil, arxiv:1710.06860 Valence Bonds in Random Quantum
More informationImprovement of Monte Carlo estimates with covariance-optimized finite-size scaling at fixed phenomenological coupling
Improvement of Monte Carlo estimates with covariance-optimized finite-size scaling at fixed phenomenological coupling Francesco Parisen Toldin Max Planck Institute for Physics of Complex Systems Dresden
More informationimmune response, spin glasses and, 174, 228 Ising model, definition of, 79-80
Index Amit, D., 135-136 Anderson localization, 49-50, 245n9 Anderson, Philip, 5, 49, 79, 127-129, 155, 169-170, 223, 225-228, 248n2, 251n2 Anfinsen, C.B., 163 annealing, 119, 245n7 simulated, see simulated
More informationarxiv:cond-mat/ v3 16 Sep 2003
replica equivalence in the edwards-anderson model arxiv:cond-mat/0302500 v3 16 Sep 2003 Pierluigi Contucci Dipartimento di Matematica Università di Bologna, 40127 Bologna, Italy e-mail: contucci@dm.unibo.it
More informationClusters and Percolation
Chapter 6 Clusters and Percolation c 2012 by W. Klein, Harvey Gould, and Jan Tobochnik 5 November 2012 6.1 Introduction In this chapter we continue our investigation of nucleation near the spinodal. We
More informationPERSPECTIVES ON SPIN GLASSES
PERSPECTIVES ON SPIN GLASSES Presenting and developing the theory of spin glasses as a prototype for complex systems, this book is a rigorous and up-to-date introduction to their properties. The book combines
More informationConference on Superconductor-Insulator Transitions May 2009
2035-10 Conference on Superconductor-Insulator Transitions 18-23 May 2009 Phase transitions in strongly disordered magnets and superconductors on Bethe lattice L. Ioffe Rutgers, the State University of
More informationInvaded cluster dynamics for frustrated models
PHYSICAL REVIEW E VOLUME 57, NUMBER 1 JANUARY 1998 Invaded cluster dynamics for frustrated models Giancarlo Franzese, 1, * Vittorio Cataudella, 1, * and Antonio Coniglio 1,2, * 1 INFM, Unità di Napoli,
More informationFrustrated diamond lattice antiferromagnets
Frustrated diamond lattice antiferromagnets ason Alicea (Caltech) Doron Bergman (Yale) Leon Balents (UCSB) Emanuel Gull (ETH Zurich) Simon Trebst (Station Q) Bergman et al., Nature Physics 3, 487 (007).
More informationMagnetism at finite temperature: molecular field, phase transitions
Magnetism at finite temperature: molecular field, phase transitions -The Heisenberg model in molecular field approximation: ferro, antiferromagnetism. Ordering temperature; thermodynamics - Mean field
More informationStudy of Condensed Matter Systems with Monte Carlo Simulation on Heterogeneous Computing Systems
Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2016 Study of Condensed Matter Systems with Monte Carlo Simulation on Heterogeneous Computing Systems Sheng Feng
More informationQuantum annealing for problems with ground-state degeneracy
Proceedings of the International Workshop on Statistical-Mechanical Informatics September 14 17, 2008, Sendai, Japan Quantum annealing for problems with ground-state degeneracy Yoshiki Matsuda 1, Hidetoshi
More informationarxiv:cond-mat/ v1 [cond-mat.stat-mech] 24 Jul 2001
Autocorrelation functions in 3D Fully Frustrated Systems arxiv:cond-mat/0107509v1 [cond-mat.stat-mech] 24 Jul 2001 G. Franzese a, A. Fierro a, A. De Candia a and A. Coniglio a,b Dipartimento di Scienze
More informationPhysics 212: Statistical mechanics II Lecture XI
Physics 212: Statistical mechanics II Lecture XI The main result of the last lecture was a calculation of the averaged magnetization in mean-field theory in Fourier space when the spin at the origin is
More informationInformation, Physics, and Computation
Information, Physics, and Computation Marc Mezard Laboratoire de Physique Thdorique et Moales Statistiques, CNRS, and Universit y Paris Sud Andrea Montanari Department of Electrical Engineering and Department
More informationJanus: FPGA Based System for Scientific Computing Filippo Mantovani
Janus: FPGA Based System for Scientific Computing Filippo Mantovani Physics Department Università degli Studi di Ferrara Ferrara, 28/09/2009 Overview: 1. The physical problem: - Ising model and Spin Glass
More informationarxiv:cond-mat/ v1 13 May 1999
Numerical signs for a transition in the 2d Random Field Ising Model at T = 0 arxiv:cond-mat/9905188v1 13 May 1999 Carlos Frontera and Eduard Vives Departament d Estructura i Constituents de la Matèria,
More informationSolving the sign problem for a class of frustrated antiferromagnets
Solving the sign problem for a class of frustrated antiferromagnets Fabien Alet Laboratoire de Physique Théorique Toulouse with : Kedar Damle (TIFR Mumbai), Sumiran Pujari (Toulouse Kentucky TIFR Mumbai)
More informationarxiv:cond-mat/ v1 [cond-mat.dis-nn] 12 Jun 2003
arxiv:cond-mat/0306326v1 [cond-mat.dis-nn] 12 Jun 2003 CONVEX REPLICA SIMMETRY BREAKING FROM POSITIVITY AND THERMODYNAMIC LIMIT Pierluigi Contucci, Sandro Graffi Dipartimento di Matematica Università di
More informationStatistics and Quantum Computing
Statistics and Quantum Computing Yazhen Wang Department of Statistics University of Wisconsin-Madison http://www.stat.wisc.edu/ yzwang Workshop on Quantum Computing and Its Application George Washington
More informationarxiv:cond-mat/ v1 [cond-mat.dis-nn] 10 Dec 1996
The Metastate Approach to Thermodynamic Chaos arxiv:cond-mat/9612097v1 [cond-mat.dis-nn] 10 Dec 1996 C.M. Newman Courant Institute of Mathematical Sciences, New York University, New York, NY 10012 D.L.
More information