AIM, Palo Alto, August 2006

T. Witten, University of Chicago

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Elements of a phase transition

• a set of N variables, e.g. "spins" {si} = 0 or 1: any particular choice is a configuration

• a list or graph of connections between variablese.g. square lattice with neighbors connected

• statistical weight for each configuration

eg "each connected pair of unlike spins

reduces weight by factor x"• a rule for joining joining N systems by adding connections

eg. enlarge lattice.

Such a system may have a phase transition:

• averages change non-smoothly with weightseg d s

dx x = xcN

s

x0 1

Non-smoothness requires N ––emergent

Phase 1 Phase 2

Variables are influenced by an infinite number of others

0

1

1 1

1

1

11

11

1

1

10

0 0

0

0 0

0

0

0

0 0

0 1

Equilibrium phase transitions

–T. Witten, University of Chicago

• varieties of phase transition

• nature of the transition state

• theoretical methods

• questions for workshop

• distinctions: two non-phase transitions

Mean field theory

Conformal symmetry

Renormalization theory

Stochastic Loewner evolution

Complex emergent structure with mysterious regularities

Phase transition versus ?

s

x0 1

• non abrupt: crossover

• non-emergent: abrupt without N

Ising model: paradigm phase transition

Probabilities P{s} are set by a "goal function" H{s}

Goal: all s's are the same as their connected neighbors

Specifically: H{s} = -J (sc1 - 1/2)(sc2 - 1/2) connections

c=1

L

0

1

1 1

1

1

11

11

1

1

10

0 0

0

0 0

0

0

0

0 0

0 1

Unlike neighbors increased H decreased probability P:

P{s} = (constant) e-J/2 (= x) if sc1 sc21 if sc1= sc2`connections

c

Has a phase transition with discontinuous derivitived sdx

… at x = xc N (and

L)

Generalizing Ising model

Variables: Z2 (Ising), Zn (clock), O2 (superfluid helium) On (magnets) …In general the variables are group elements and the goal

function depends on the group operation relating the connected elements

Graph: d-dimensional lattice, continuous space, Cayley tree …

connections beyond nearest-neighbor

connections among k > 2 variables

Goal function: seeks like values of variables / seeks different valuesSatisfiable in finite fraction, vanishing fraction

(Ising), or none (frustrated) of the configurations

Probabilities: thermal equilibrium (Ising): each graph element contributes one

factor to the probability; factor depends on goal function on that element.

kinetic: configurations are generated by a sequential, stochastic process whose probabilities are dictated by a goal functionStochastic growth processes

Definite / random: eg, each connection has weight x or 1/x chosen

randomly: spin glass

"nearly empty lattice" generalizations

Self-avoiding random walk

Random animals

percolation

Goal function allows only sites that form a linear sequence from the origin.

Penalty factor x for each site

Goal function allows only cluster of connected sites

Penalty x for each site

Place s variables on a lattice at random with a given s = x

Determine largest connected cluster

All have phase transitions: as x xc , size of cluster grows to a nonzero fraction of the lattice

Varieties of phase-transition behavior

How non-smooth?

The transition state

discontinuous

continuous

heterogeneous

State depends on history, boundary conditions

Jochen Voss: http://seehuhn.de/mathe/ising-0.219.jpg

slow: correlation time diverges at xc

complicated

Discontinuous: expectation values eg s jump at transition point eg boiling

Continuous: only derivitives of s are discontinuous: eg Ising model, critical opalescence

aka first-order, subcritical

aka second-, third- …order, critical

Uniform regions: eg liquid and vapor

System is uniform over length scales >"correlation length"

x xc

Character of heterogeneity: dilation invarianceDilated configurations are indistinguishable

Eg random walks with steps much smaller than resolution

4 of these are zoomed 3x relative to the other four. Can you tell which?

s(0) s(r1)s(r2)…s(rk)

= -Ak s(0) s(r1)s(r2)…

s(rk)

Dilation invariance means that statistical averages are virtually unchanged by dilation:

A is the "scaling dimension" of s

This invariance appears to hold generally for continuous phase transitions

A governs response of length to a change of x

~ (x - xc)-

1/(d-A)A set of "critical exponents" like A dictate response near the transition point

Character of the transition state: universality

"Relevant" features do change the exponents:

Discrete universality classes

Spatial dimension d

Critical exponents like A are often invariant under continuous changes of the system

Eg Ising model on different lattices, different variables, different goal functions

Eg liquid vapor, ferromagnet, liquid demixing, ising

These changes are called "irrelevant variables"

Symmetry of the system: Z2 (Ising),

O2, … O3

Theoretical understanding of phase transitions

• Mean-field theory• Renormalization theory• Conformal symmetry• Stochastic Loewner evolution• (Defects (local departures from goal

state) ….omitted)• (replica methods, omitted)

Mean-field approximation: neglect nonuniformity

In H(s), replace spins connected to sI by their average s

This converts system to a single degree of freedom sI

One can readily compute eg thermodynamic free energy F(x,s) for assumed s

The actual s is that which minimizes free energy.

Mean field theory accounts for phase transitions.

0.2 0.4 0.6 0.8 1

-0.68

-0.66

-0.64

-0.62

-0.6

-0.58

J = .45 J = .55

free energy at temperature = 1

magnetization M

energy

p p

entropy

Inherently independent of space dimension d

Average s "magnetization" M takes the value that minimizes free energy

Renormalization theory

To explain dilation symmetry requires a dilation-symmetric description

To describe large-scale structure, we may remove small-scale structure of {s} by averaging over distance >> graph connections to produce s(r)

If H[s] describes the system and s is dilation symmetric, then

H2 must have the same functional form as H

Eg for d-dimensional Ising model with d 4 at transition

H ( s)2 +

g*(d) s4

One infers the mapping H H2 from the weights P[s2]:

H P[s]

P[s2] H2Local spatial averaging: 2

From the transformation of H near transition point, one

infers the effect of dilation: "scaling dimension" A of s other critical exponents.

––Wilson, Fisher 1972

Many H's converge to

same H: explains

universality

Conformal symmetry for 2-dimensional systems

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configuration in z plane g(z) shows deformedconfiguration

Any analytic functiong(z)

Conformal symmetry: deformed configuration has same statistical weight as original.

–Shankar, Friedan ~1980

Critical exponents label a discrete set of representations of conformal groupCf discrete angular momentum representations of rotation symmetry

Restricted to d=2

Loewner: analytic map implements self-avoidance

Cf. John Cardy cond-mat/0503313

Zg(z) = a +

((z-a)2 + 4t)1/2 g

a a

√t)

Any self-avoiding curve has a g(z), analytic in

Growing curve: tip z0 came from some point at on the real axis.

z0

By varying at with time, we can make “arbitrary” self avoiding curves.

Loewner: affect of at on gt(z) is local in time:

Note: if at is held fixed for t, z0 “diffuses upward” with diffusion constant 4

Each point on the current curve gt(z) feels Coulomb repulsion from at.

Schramm: Loewner growth with Brownian at

suppose at diffuses, with ‹(at)2 › = tCurve is dilation symmetric (fractal) with fractal dimension D = 1 - /8

Many known random lattice structures are Schramm curves, differing only in

Self-avoiding walk: = 8/3

Ising cluster perimeters: = 3Percolation cluster boundaries: = 6

…

Brownian motion replaces field theory to explain many known universal dilation-symmetric structures.

… simple random walks can create a new range of “correlated” objects.

Generalizing the notion of phase transitions

Processes in this workshop (eg k-sat) act like phase transitions

Sharp change of behavior

Emergent structureScale invariance

complexity

Without the features thought fundamental in conventional phase transition

Statistical mechanics description

Spatial dimensionality.

Can we free phase transitions from these features to get a deeper understanding?

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Correlation qualitatively alters structure

Random walk: no correlation Self-avoiding walk: correlation

Size ~ length1/2 Size ~ length3/4

central limit theorem Strong modifications needed to defeat central limit theorem.

…interacting field theory3/4 exponent was only known by

simulationconjectures

Future hopes for Schramm-Loewner (SLE)

Can it be generalized to account for branched objects, eg. Random animals

Can it describe stochastic growth phenomena, eg. DLA?

Can it be generalized to higher dimensions?

SLE shows that many known dilation-symmetric structures are simple kinetic, random walks, viewed through the distorting lens of evolving maps

Random structures like self avoiding walks are universal fractals in any dimension

If they are Schramm-like in two-dimensions, why not in others?What plays the role of gt(z)?

Displacement field of mapping a region into itself

Must preserve topology

Must preserve dilation symmetry.

Must have a singular point: growing tip.