The ultrametric tree of states and computation of ... · Andrea Lucarelli – The ultrametric tree...

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 functions in spin glasses

Andrea Lucarelli

Summary

•  Introduction •  Broken symmetries and Goldstone bosons •  The replica approach •  Toy model •  Full theory •  Diagrammatic expansion •  Fat diagrams •  Bethe approximation •  Correlation functions •  Conclusions

Andrea Lucarelli – The ultrametric tree of states and computation of correlation functions in spin glasses

Disordered systems: spin glasses

Biology

Proteins

Neural networks

River basins morphology

Keyword: collective behavior of a large heterogeneous system of interacting agents

Networks

Finance networks

Evolution networks

Internet networks


Ising spin glass hamiltonian: symmetry and symmetry breaking…

H[{s}]Ising = -∑JijSiSj - h∑iSi, Si=±1, i=1,…,N

quenched parameters Jij Gaussian distribution zero average variance J2=1/N

magnetic field h nearest neighbours

the energy of a state{si} is precisely the same as the energy of the state with every spin flipped {-si}

with h≠0 the symmetry is explicitly broken: the Hamiltonian does not have the s→−s symmetry (Z2).


True beauty is a deliberate, partial breaking of symmetry (Zen proverb)

Ising spin glass hamiltonian: symmetry and symmetry breaking…

H[{s}]Ising = -∑JijSiSj, Si=±1, i=1,…,N

alternative definition (in the continuum)

We obtain the previous definition when the gauge group G is Z2, we are on the lattice and we consider the strong coupling limit.

HA[{g}] = ∫dxTr(Aµg(x))2 gauge group G → Z2 gauge field Aµ(x) → J

gauge transform g(x) → σ

In many cases Gribov ambiguity tells us that HA(g) has many minima, therefore HJ (σ ) has an exponentially large number of minima.


Ising spin glass hamiltonian


Finding minimum

energy configuration

given Jij

Si= +1 Si= -1


Ising spin glass hamiltonian


states unrelated to one another by simple symmetries and separated by very high free-energy barriers.

Local magnetization for each state miα = ⟨σi⟩α Distance qαβ = 1/N ∑i miα miβ


For T

Scalar field and broken symmetries

At high temperature, no symmetry breaking At low temperature, the scalar field symmetry is broken There are two contributions: •  longitudinal direction •  transverse direction


Order parameter ≠0, Q=Q+∂Q

Higgs particles are quantum excitations (ripples) of the Higgs field.

Quantum excitations which push along the circle are called Goldstone bosons.

1-D version of the Higgs potential Higgs potential over the complex plane

Correlation functions and divergencies

For D=4 theory, e.g. a theory with interaction φ4: 1/k2 propagator possible IR singularities k= 0 vertex is canceled, IR singularities are reduced.


Computation of the corrections to mean field theory of Spin Glass lead to very complicated bare propagators, with severe IR divergencies. •  What is their structure? •  Can they be somehow reduced?

When a continuous symmetry is spontaneously broken, new massless scalar particles appear in the spectrum of possible excitations (Goldstone theorem).

4 Goldstone bosons scattering

The replica approach

H[{s}]Ising = -∑JijSiSj - h∑iSi, Si=±1, i=1,…,N

Free energy density in powers of Q

Functional in terms of q [0,1]

Stationarity equations wr to q for Ta< si >b between two states a and b differing by a finite amount in free energy.


Replica approach (effective theory): average over the disorder the partition function of n copies (replicas) of the original model, n being analytically continued to zero at the end.

Overlaps and the ultrametric tree

The overlap these states are organized ultrametrically. By putting the states at the end of the branches of a tree, the overlap between the states can be represented by the distance between the top root and the level of the point where the branches coincide. Given three states at least two overlaps are equal,

the third being greater than the other two


Longitudinal, anomalous, replicon

Order parameter Q Fluctuations around the RSB saddle point

the fluctuations of the order parameter Q around the RSB saddle point are usually divided into three families

Anomalous

invariant under the permutations symmetry of the n replicas break even this n replica permutation group

L

R

A


Projected propagator

small conjugate field ε (explicit RSB)

Bare propagator G(x,y;p)

an explicit RSB can be introduced in the theory by adding to the effective free energy the term

kinetic term


for small p there is an off-diagonal contribution of width p and order p−3 that cancels the p−2 singularity and leads to a massive propagator.

x=y

Green functions and divergencies

Propagator 2° order

Disconnected diagram 2° order

Longitudinal Anomalous GF

Replicon GF

O(p-3) divergences u-1 ultrametric prefactor O(p-2) divergences x-2 ultrametric prefactor

Propagator (mass matrix with diag kinetic term)-1


u= min(x,y) v= max(x,y)

Ultrametric trees 4 ultrametric indices → different possibilities of arranging them on an ultrametric tree


Different topologies xab = distance between a and b

G studied in all phase space (# terms ~ 106) The propagator G has a strange effect in some corners of the phase space; there is a large number of cancellations (due to the Goldstone boson)

Taxonomic structure of the tree of states (K=3) Iterative generation of a pruned tree with K=3 RSB


Diagrammatic expansion


3. Random magnetic Ising ferromagnets at

T=0

Esperimenti di alta precisione (a bassa

energia) [frontiera di alta

intensità]

Bethe mean field theory vs mean field theory

1. Random quantum Hamiltonian

Esperimenti ad alta energia

[frontiera di alta energia]

2. T=0 Spin Glass in a magnetic field

Esperimenti di alta precisione (a bassa

energia) [frontiera di alta

intensità]

Bethe mean field Mean field approx

the transition from localized to extended states for a random

quantum Hamiltonian.

Localized states do not exist in the usual mean field

approximation the transition from the glassy

phase to the paramagnetic phase in zero temperature spin

glasses in a magnetic field.

In mean field approximation at T=0 we are always in the

glassy phase

in the study of random magnetic Ising ferromagnets at zero temperature there is a

region where many extensively different local

minima do exist

This does not exist in the standard mean field theory approach.


Diagrammatical expansion from Bethe MFT and not from the naïve MFT.

Fat diagrams: expansions around mean field theory

Perturbation around the Bethe approximation correspond to a new kind of diagrams: fat diagrams. This approach was introduced in the ’90 by Efetov, later was studied by Parisi and Slanina. Why? ü  in the other cases standard diagrammatic expansion doesn’t exist ü  fat diagrams similar to the old ones (with a simpler interpretation). Different computation rules ü  fat diagrams approach may simplify the computation when there are many cancellations.


MFT exact solutions are difficult, so it is convenient to study the perturbative expansion around the MF. Ising model on lattice

Bethe approximation

•  Let’s consider point i •  Let’s call Vi the set of the points k such that Ai,k=1 (the neighbourhood of i). •  Let’s remove the spin i •  Let’s suppose that in this situation the spin in Vi are uncorrelated (they must be

correlated when the spin i is present) cavity magnetizations mC (self consistent equation for these magnetizations) a Bethe lattice is a graph where the Bethe approximation is exact, e.g. Random Regular Graph RRG with coordination number z. When N →∞ RRG is locally a tree: the probability to find a loop of length L containing a point goes to zero as N-1(z-1)L. Typical loops have a lenght of order log (N) The Bethe approximation is correct on RRG for N →∞


Bethe lattice


A Bethe lattice (Bethe, 1935), is an infinite connected cycle-free graph where each node is connected to z neighbours, where z is called the coordination number.

A tree is a network in which there are no loops. When one of the node is removed, the tree will be split into two tree. Bethe lattice is an infinite tree, and effective d=∞ lattice.

M-layered model

Infinite range M-layered model Short range M-layered model

NxM spins sia i=1,…,N a=1,…,M NxM spins sia i=1,…,N a=1,…,M The Hamiltonian is

The Hamiltonian is

where for each edge i,k we have a quenched permutation pa(i,k) of M elements. For M=1, we recover the original model.

If M→∞we have the saddle point equation

The 1/M expansion is the standard loop expansion

For each point i the number of closed loops of length L starting and ending in i,a is If M→∞ at fixed L the saddle point equation and the Bethe approximation is exact.


M-layer model

In the limit M -> ∞ we get a random graph Bethe approximation is exact (Vontobel, 2012) d-dimensional regular lattice -> 2d-regular random graph Typical loops are of order O(M)

M=∞ random regular lattice, i.e. locally a tree (genus 0) Bethe, no loops M finite: there are loops •  with probability 1/M: just one loop •  with probability 1/M2 two loops M=1 standard lattice, usual D-dimensional lattice: all genera

1/M = loop expansion 1/Mk =genus k expansion


1/M expansion around Bethe solution

0-‐th order: 0 loops = number of RW at 9me L and posi9on x 1-‐st order: 1 loop


Linear and Loop (Bethe) correlations at T=0

Linear Bethe computation at T=0 1 loop Bethe computation at T=0

S1 S2 S3

h h

S1 S3

è

S1 S2 S3 S4 è

S1 S4

L2 L1 L4 L3


Bethe approx in a d-dimensional lattice

In order to describe the approach it is convenient to recall the particle representation in field theory. In D dimensions the free propagator 1-loop contribution


probability of a path of length (internal time) s going from 0 to x.

Bethe approximation can be defined only for a theory on the lattice. The typical Hamiltonian is of the form: where the sum is done over the nearest neighbor points.

Bethe approx in a d-dimensional lattice

Probability distribution of the field φ : Bethe approximation B(φ) the probability distribution the field with z − 1 neighbours (we remove one spin and we consider the probabability distribution of the spins around this cavity). Correlation functions: unique path from point x to y of lenght L The effects of the loops can be included by considering the contribution of lattice region with high genus. New objects (fat diagrams): they look like usual diagrams, but they contain the resummation of non-perturbative effects in one dimensions.


Computing correlations functions

We are interested in the correlation functions for a Spin Glass with field on a Bethe lattice, in the high connectivity limit (z→ ∞). In this limit in fact things are easier. Couplings: Higher orders are negligible in z→ ∞ limit. replicated partition function expansion


from where we can extract connected and disconnected correlation function

Tab,cd=QabQcd

Eigenvalues and projectors on eigenvalues

Anomalous

L

R

A


(multiplicity n(n-3)/2)

(multiplicity n-1)

We are interested in the correlations at distance k, so we want to compute Tk:

in the limit n→0 λ1=λ2 (so the first addend can be rewritten)

Connected and disconnected functions

•  Ising spins •  SG in field •  RFIM

Zero temperature ferromagnes in a random magnetic field


Disconnected function

magnetizations (mean value)


two spins Hamiltonian At T=0 we have the ground state σ1*, σ2* of the Hamiltonian. We can group the result of minimization in three different scenarios.

Disconnected function


two spins Hamiltonian

Assumiamo che c, n1, n2, k1, k2 ~O(1) siano quantità finite. Ansatz per la funzione di correlazione

Conclusions and perspectives


•  comparison between analytical results and simulations

Ferromagnetic random magnetic systems at T=0

•  the one dimensional study is the same of the previous case

Spin glass random magnetic systems at T=0

•  starting level (1/N correction to the spectral density of the Lagrangian on RRG)

Localization

•  first steps

Jamming

What next So far…

Jamming

Localization

SG at T=0

Expansion around Bethe

•  finite z • RSB

Expansion around Bethe solution

To do:

Università degli studi di Roma “La Sapienza”

The ultrametric tree of states and

computation of correlation functions in spin glasses

Andrea Lucarelli

The ultrametric tree of states and computation of ... · Andrea Lucarelli – The ultrametric tree...

Documents

Transcript of The ultrametric tree of states and computation of ... · Andrea Lucarelli – The ultrametric tree...