magnetization field at criticality in the ising model
play

Magnetization field at criticality in the Ising model Christophe - PowerPoint PPT Presentation

Nov 17, 2022 •201 likes •1.05k views

Magnetization field at criticality in the Ising model Christophe Garban ENS Lyon and CNRS Federico Camia (Vrije Universiteit Amsterdam) Joint work with Charles Newman (NYU) and Percolation and Interacting Systems, MSRI, February 2012 C. Garban

  1. Magnetization field at criticality in the Ising model Christophe Garban ENS Lyon and CNRS Federico Camia (Vrije Universiteit Amsterdam) Joint work with Charles Newman (NYU) and Percolation and Interacting Systems, MSRI, February 2012 C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 1 / 25

  2. C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 2 / 25

  3. C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 2 / 25

  4. What are the fluctuations of the magnetiza- tion field ? C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 2 / 25

  5. N N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 3 / 25

  6. + + − − − − − − + + + + + − − − + + − − − − − − + + − + + + − + + + − − − − + + − + − − + − − − + + − + N − − − − + + + − + − − − + + − + N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 3 / 25

  7. • If the spins are i.i.d � σ x → N (0 , 1) 1 N + + − − − − − − + + + + + − − − + + − − − − − − + + − + + + − + + + − − − − + + − + − − + − − − + + − + N − − − − + + + − + − − − + + − + N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 3 / 25

  8. • If the spins are i.i.d � σ x → N (0 , 1) 1 N � σ x δ x 1 → W , two- + + − − − − − − N + + + + + − − − dimensional white noise + + − − − − − − + + − + + + − + + + − − − − + + − + − − + − − − + + − + N − − − − + + + − + − − − + + − + N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 3 / 25

  9. + + − − − − − − + + + + + − − − + + − − − − − − + + − + + + − + + + − − − − + + − + − − + − − − + + − + N − − − − + + + − + − − − + + − + N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 4 / 25

  10. Ising model To each configuration σ ∈ {− 1 , 1 } N 2 , one + + associates the Hamiltonian − − − − − − + + + + + − − − + + − − − − − − + + − + + + − + + + − − − − H ( σ ) := − � i ∼ j σ i σ j + + − + − − + − − − + + − + N − − − − + + + − + − − − + + − + N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 4 / 25

  11. Ising model To each configuration σ ∈ {− 1 , 1 } N 2 , one + + associates the Hamiltonian − − − − − − + + + + + − − − + + − − − − − − + + − + + + − + + + − − − − H ( σ ) := − � i ∼ j σ i σ j + + − + − − + − − − + + − + N − − − − + + + − And we define: + − − − + + − + P β ( σ ) ∝ e − β H ( σ ) N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 4 / 25

  12. • If β < β c � σ x → N (0 , σ 2 1 β ) N � σ x δ x → σ β W 1 + + − − − − − − N + + + + + − − − + + − − − − − − + + − + + + − + + + − − − − + + − + − − + − − − + + − + N − − − − + + + − + − − − + + − + N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 5 / 25

  13. • If β < β c � σ x → N (0 , σ 2 1 β ) N � σ x δ x → σ β W 1 + + − − − − − − N + + + + + − − − + + − − − − − − + + − + + + − + • If β > β c + + − − − − + + − + − − + − − − + + − + N − − − − + + + − + − − − + + − + N N 2 � σ 0 � − N 2 � σ 0 � + C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 5 / 25

  14. • If β < β c ⊕ � σ x → N (0 , σ 2 1 β ) N � σ x δ x → σ β W 1 + + − − − − − − N + + + + + − − − + + − − − − − − + + − + + + − + • If β > β c + + − − − − + + − + − − + − − − � σ x − N 2 E + + − + � � σ 0 N − − − − → N (0 , σ 2 β ) + + + − N + − − − + + − + • As β → β c , σ β ր ∞ N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 5 / 25

  15. • If β < β c ⊕ � σ x → N (0 , σ 2 1 β ) N � σ x δ x → σ β W 1 + + − − − − − − N + + + + + − − − + + − − − − − − + + − + + + − + • If β > β c + + − − − − + + − + − − + − − − � σ x − N 2 E + + − + � � σ 0 N − − − − → N (0 , σ 2 β ) + + + − N + − − − + + − + • As β → β c , σ β ր ∞ N And now: what about β = β c ?? C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 5 / 25

  16. Which normalization ? To avoid boundary issues, consider our system on the torus Z 2 / N Z 2 . We have: � � � � Var ( σ x ) = σ x σ y E x , y

  17. Which normalization ? To avoid boundary issues, consider our system on the torus Z 2 / N Z 2 . We have: � � � � Var ( σ x ) = σ x σ y E x , y N 2 � � � ≍ σ 0 σ y E y

  18. Which normalization ? To avoid boundary issues, consider our system on the torus Z 2 / N Z 2 . We have: � � � � Var ( σ x ) = σ x σ y E x , y N 2 � � � ≍ σ 0 σ y E y | y | − 1 / 4 (by Onsager) N 2 � ≍ y N 2 N 2 N − 1 / 4 = N 15 ≍ 4

  19. Which normalization ? To avoid boundary issues, consider our system on the torus Z 2 / N Z 2 . We have: � � � � Var ( σ x ) = σ x σ y E x , y N 2 � � � ≍ σ 0 σ y E y | y | − 1 / 4 (by Onsager) N 2 � ≍ y N 2 N 2 N − 1 / 4 = N 15 ≍ 4 � σ x Hence it is natural to look at the random variable m ( N ) := N 15 / 8 . Question Does m ( N ) have a ( unique ) scaling limit ?

  20. “Subtle” issue of renormalization Theorem (McCoy, Wu, 1967) As N → ∞ ∼ c N − 1 / 4 � � σ 0 , 0 σ N , N where c = 2 1 / 12 e 3 ζ ′ ( − 1 ) C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 7 / 25

  21. “Subtle” issue of renormalization Theorem (McCoy, Wu, 1967) As N → ∞ ∼ c N − 1 / 4 � � σ 0 , 0 σ N , N where c = 2 1 / 12 e 3 ζ ′ ( − 1 ) Proposition ( Rotational invariance of the two-point function) � � σ 0 σ x − → � x � 2 →∞ 1 � � σ 0 σ � x � 2 C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 7 / 25

  22. Notations / definitions Rescaled lattice a Z 2 , a ≪ 1. Definition (Renormalized magnetization field) + + − − − − − − + + + + + − − − + + − − − − − − Φ a := 15 + + − + + + − + � δ x σ x a 8 + + − − − − + + − + − − x ∈ a Z 2 + − − − + + − + m a := � Φ a , 1 [ 0 , 1 ] 2 � − − − − + + + − + − − − + + − + a Z 2 m a L := � Φ a , 1 [ 0 , L ] 2 � 1

  23. Notations / definitions Rescaled lattice a Z 2 , a ≪ 1. Definition (Renormalized magnetization field) + + − − − − − − + + + + + − − − + + − − − − − − Φ a := 15 + + − + + + − + � δ x σ x a 8 + + − − − − + + − + − − x ∈ a Z 2 + − − − + + − + m a := � Φ a , 1 [ 0 , 1 ] 2 � − − − − + + + − + − − − + + − + a Z 2 m a L := � Φ a , 1 [ 0 , L ] 2 � 1 Question The field Φ a ∈ D ′ . Is it the case that Φ a converges as a → 0 to some random distribution Φ ?

  24. Road map • First part : Theorem (Camia, G., Newman) (i) The magnetization field Φ a on a Z 2 has a unique scaling limit as the mesh a → 0 .

  25. Road map • First part : Theorem (Camia, G., Newman) (i) The magnetization field Φ a on a Z 2 has a unique scaling limit as the mesh a → 0 . (ii) On a finite domain Ω with boundary conditions ξ , the magnetization field | Ω ∩ a Z 2 has a scaling limit Φ ξ Φ a Ω .

  26. Road map • First part : Theorem (Camia, G., Newman) (i) The magnetization field Φ a on a Z 2 has a unique scaling limit as the mesh a → 0 . (ii) On a finite domain Ω with boundary conditions ξ , the magnetization field | Ω ∩ a Z 2 has a scaling limit Φ ξ Φ a Ω . (iii) In particular, m a has a unique limiting law m = m ξ which depends on the boundary conditions (usually + , − , or free ).

  27. Road map • First part : Theorem (Camia, G., Newman) (i) The magnetization field Φ a on a Z 2 has a unique scaling limit as the mesh a → 0 . (ii) On a finite domain Ω with boundary conditions ξ , the magnetization field | Ω ∩ a Z 2 has a scaling limit Φ ξ Φ a Ω . (iii) In particular, m a has a unique limiting law m = m ξ which depends on the boundary conditions (usually + , − , or free ). (iv) The scaling limit is NOT Gaussian.

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Planar Ising model at criticality: state-of-the-art and perspectives Dmitry Chelkak, ENS,

Planar Ising model at criticality: state-of-the-art and perspectives Dmitry Chelkak, ENS,

Planar Ising model at criticality: state-of-the-art and perspectives Dmitry Chelkak, ENS, Paris [ &amp; PDMI, St. Petersburg (on leave) ] ICM, Rio de Janeiro, August 4, 2018 Planar Ising model at criticality: outline Combinatorics

791 views • 37 slides
2D Ising Model: Near-Critical Scaling Limit and Magnetization Critical Exponent Charles M.

2D Ising Model: Near-Critical Scaling Limit and Magnetization Critical Exponent Charles M.

2D Ising Model: Near-Critical Scaling Limit and Magnetization Critical Exponent Charles M. Newman Courant Institute of Mathematical Sciences newman @ courant.nyu.edu Based on joint work with Federico Camia and Christophe Garban. Ising

604 views • 23 slides
A power-law upper bound on the correlations in the two-dimensional random-field Ising model Ron

A power-law upper bound on the correlations in the two-dimensional random-field Ising model Ron

A power-law upper bound on the correlations in the two-dimensional random-field Ising model Ron Peled, Tel Aviv University Based on a joint work with Michael Aizenman, Princeton ICMP 2018, Montreal Random-field Ising model Standard Ising

447 views • 12 slides
The Ising Spin Lattice Model Alexander Brunmayr Graduate mentor: Nicholas Paskal &amp; Stephen

The Ising Spin Lattice Model Alexander Brunmayr Graduate mentor: Nicholas Paskal &amp; Stephen

The Ising Spin Lattice Model Alexander Brunmayr Graduate mentor: Nicholas Paskal &amp; Stephen Sorokanich August 28, 2018 Mini Physics Intro Course Magnets and spins, magnetization Ferromagnetism Spins under an external magnetic field Ising

286 views • 12 slides
Quenching to field-stabilized Adam Iaizzi* Postdoctoral Fellow magnetization plateaus in the

Quenching to field-stabilized Adam Iaizzi* Postdoctoral Fellow magnetization plateaus in the

J24.00013 Quenching to field-stabilized Adam Iaizzi* Postdoctoral Fellow magnetization plateaus in the National Taiwan University unfrustrated Ising antiferromagnet Better title: Stable frozen states without disorder

413 views • 15 slides
Quantum Integer Programming 47-779 Ising Model 1 William Larimer Mellon, Founder Agenda o

Quantum Integer Programming 47-779 Ising Model 1 William Larimer Mellon, Founder Agenda o

Quantum Integer Programming 47-779 Ising Model 1 William Larimer Mellon, Founder Agenda o Ising Model: Background, Physics o Ising Model: Solutions o Ising Model and Combinatorial Optimization o Ising Model and Integer Programming o Solving

669 views • 19 slides
Height representation of XOR-Ising loops via bipartite dimers C edric Boutillier (UPMC) B

Height representation of XOR-Ising loops via bipartite dimers C edric Boutillier (UPMC) B

Height representation of XOR-Ising loops via bipartite dimers C edric Boutillier (UPMC) B eatrice de Tili` ere (UPMC) LAGA Universit e Paris Nord March 11, 2015 The Ising model and the XOR-Ising model The Ising model Let G =

797 views • 43 slides
1D Ising model Simple model of interacting spins in a lattice Nearest-neighbor only No

1D Ising model Simple model of interacting spins in a lattice Nearest-neighbor only No

1D Ising model Simple model of interacting spins in a lattice Nearest-neighbor only No external magnetic field Here, we look at a 1D chain of spins Future: higher dimensions, higher coordination numbers Numerical Challenge

370 views • 4 slides
The critical Z -invariant Ising model via dimers B eatrice de Tili` ere University of

The critical Z -invariant Ising model via dimers B eatrice de Tili` ere University of

The critical Z -invariant Ising model via dimers B eatrice de Tili` ere University of Neuch atel joint work with C edric Boutillier, University Paris 6 Ascona, 27 May 2010 The 2 -dimensional Ising model Planar graph, G = ( V ( G )

600 views • 24 slides
1st PRIMA Congress Conformal invariance and universality in the 2D Ising model Stanislav Smirnov

1st PRIMA Congress Conformal invariance and universality in the 2D Ising model Stanislav Smirnov

1st PRIMA Congress Conformal invariance and universality in the 2D Ising model Stanislav Smirnov July 6, 2009 Archetypical example of a phase transition: 2D Ising model Configurations of + and spins (red and blue squares) Prob( config )

669 views • 31 slides
SLE and conformal invariance for critical Ising model Stanislav Smirnov jointly with Dmitry

SLE and conformal invariance for critical Ising model Stanislav Smirnov jointly with Dmitry

3rd LA PIETRA WEEK IN PROBABILITY Stochastic Models in Physics Firenze, June 23-27, 2008 SLE and conformal invariance for critical Ising model Stanislav Smirnov jointly with Dmitry Chelkak 1D Ising model 0 1

355 views • 20 slides
Crossing probabilities in the critical 2D Ising model Dmitry Chelkak (PDMI Steklov, St.Petersburg)

Crossing probabilities in the critical 2D Ising model Dmitry Chelkak (PDMI Steklov, St.Petersburg)

Crossing probabilities in the critical 2D Ising model Dmitry Chelkak (PDMI Steklov, St.Petersburg) joint work with Stanislav Smirnov (Geneva) arXiv:0910.2045 : Universality in the 2D Ising model and conformal invariance of fermionic

667 views • 42 slides
Monte Carlo simulation of the 2D Ising model - tutorial - Zoltn Nda Babe-Bolyai University

Monte Carlo simulation of the 2D Ising model - tutorial - Zoltn Nda Babe-Bolyai University

Monte Carlo simulation of the 2D Ising model - tutorial - Zoltn Nda Babe-Bolyai University Department of Theoretical and Computational Physics . A basic Metropolis Algorithm for simulating the 2D and 3D Ising model on square lattice

462 views • 8 slides
Ising superconductivity in gated MoS2 17.12.15

Ising superconductivity in gated MoS2 17.12.15

Ising superconductivity in gated MoS2 17.12.15 Background Ising superconductor (Zeeman-protected superconductor): Intrinsic spin-orbit coupling induced effective Zeeman field with

1.22k views • 32 slides
Ising Model and Percolation Problem Shahin Rouhani Physics Department Sharif University of

Ising Model and Percolation Problem Shahin Rouhani Physics Department Sharif University of

Statistical Mechanics of Two Dimensional Critical Curves Ising Model and Percolation Problem Shahin Rouhani Physics Department Sharif University of Technology Tehran, Iran. Scale invariance in the critical Ising model Let us illustrate

394 views • 27 slides
Convergence of correlations in the 2D Ising model: primary fields [ and the stress-energy tensor

Convergence of correlations in the 2D Ising model: primary fields [ and the stress-energy tensor

Convergence of correlations in the 2D Ising model: primary fields [ and the stress-energy tensor ] Dmitry Chelkak [ ENS, Paris &amp; Steklov Institute, St. Petersburg ] [ Sample of a critical 2D Ising configuration (with two disorders), c

1.24k views • 54 slides
Near-critical Ising model Christophe Garban ENS Lyon and CNRS 8th World Congress in Probability

Near-critical Ising model Christophe Garban ENS Lyon and CNRS 8th World Congress in Probability

Near-critical Ising model Christophe Garban ENS Lyon and CNRS 8th World Congress in Probability and Statistics Istanbul, July 2012 C. Garban (ENS Lyon and CNRS) Near-critical Ising model 1 / 19 Plan 1 Near-critical behavior, case of

778 views • 76 slides
Boreas Fridge Red B American families throw away 25% of their food, worth $2,275 per year. 1 The

Boreas Fridge Red B American families throw away 25% of their food, worth $2,275 per year. 1 The

Boreas Fridge Red B American families throw away 25% of their food, worth $2,275 per year. 1 The Boreas Fridge kills microbes to keep food fresh Key Features Low Pressure/ Air Ionizer Copper Handle Humidity/O2 Slow pathogen Kill airborne

211 views • 10 slides
Solving the Helmholtz equation via row- projections Tristan van Leeuwen (Univ. of

Solving the Helmholtz equation via row- projections Tristan van Leeuwen (Univ. of

Solving the Helmholtz equation via row- projections Tristan van Leeuwen (Univ. of BC, Canada) Dan Gordon (Univ. of Haifa, Israel) Rachel Gordon (Technion, Israel) Felix J. Herrmann

423 views • 31 slides
7th focused semester on Quantum Groups E. Germain, R. Vergnioux GDR Noncommutative Geometry,

7th focused semester on Quantum Groups E. Germain, R. Vergnioux GDR Noncommutative Geometry,

7th focused semester on Quantum Groups E. Germain, R. Vergnioux GDR Noncommutative Geometry, France July 2d, 2010 E. Germain, R. Vergnioux (EU-NCG RNT) 7th focused semester on Quantum Groups July 2d, 2010 1 / 17 Quantum groups and

460 views • 28 slides
Imaginary chaos Janne Junnila (EPFL) Les Diablerets, February 10th 2020 joint work with Eero

Imaginary chaos Janne Junnila (EPFL) Les Diablerets, February 10th 2020 joint work with Eero

Imaginary chaos Janne Junnila (EPFL) Les Diablerets, February 10th 2020 joint work with Eero Saksman and Christian Webb; Juhan Aru, Guillaume Bavarez and Antoine Jego 1 Outline 1. Introduction 2. Moments 3. XOR-Ising model 4. Regularity,

985 views • 67 slides
Convergence to SLE 6 for Percolation Models (joint with I. Binder &amp; L. Chayes) Helen K. Lei

Convergence to SLE 6 for Percolation Models (joint with I. Binder &amp; L. Chayes) Helen K. Lei

Convergence to SLE 6 for Percolation Models (joint with I. Binder &amp; L. Chayes) Helen K. Lei August 14, 2009 Introduction Setup &amp; Scaling Limit Conformal Invariance &amp; Cardys Formula Statement of Result Percolation

281 views • 27 slides
Conformal Invariance of the Exploration Path in 2D Critical Bond Percolation in the Square Lattice

Conformal Invariance of the Exploration Path in 2D Critical Bond Percolation in the Square Lattice

Conformal Invariance of the Exploration Path in 2D Critical Bond Percolation in the Square Lattice Phillip YAM Chinese University of Hong Kong, STAT December 12, 2012 (Joint work with Jonathan TSAI (HKU) and Wang ZHOU(NUS)) Phillip YAM

399 views • 36 slides
Cardy embedding of random planar maps Nina Holden ETH Z urich, Institute for Theoretical

Cardy embedding of random planar maps Nina Holden ETH Z urich, Institute for Theoretical

Cardy embedding of random planar maps Nina Holden ETH Z urich, Institute for Theoretical Studies Collaboration with Xin Sun. Based on our joint works with Bernardi, Garban, Gwynne, Lawler, Li, and Sep ulveda. August 1, 2019 Holden (ETH

1.09k views • 80 slides

More recommend