The ultrametric tree of states in spin glasses: perturbative …€¦ · Andrea Lucarelli – The ultrametric tree of states in spin glasses H A [{g}] = ∫dxTr(A µ g(x))2 gauge

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 in spin glasses: perturbative analysis and explicit generation

Andrea Lucarelli

Summary

•  Introduction

•  Broken symmetries and Goldstone bosons •  The replica approach

•  Toy model

•  Full theory •  Conclusions

Andrea Lucarelli – The ultrametric tree of states in spin glasses

Introduction


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).


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.

…like in the Standard Model

The gauge symmetry → structure of strong, weak and electromagnetic interactions The global flavour symmetry → three families

Both symmetries must be broken to account for the observed masses of the elementary constituents

The “Standard Model” is a highly successful mathematical model for the description of the basic constituents of matter and their fundamental interactions. It describes with success a variety of phenomena, covering a huge range of energies: from few eV (atomic energies) up to ~ 1 TeV (LHC collisions). It is a Relativistic Quantum Field Theory with two main ingredients: A set of underlying symmetries + A symmetry-breaking sector:

LSM = Lgauge(Aa,ψi) + LSymm.Break.(φ,Aa,ψi) True beauty is a deliberate, partial breaking of symmetry


Ising spin glass hamiltonian


Finding minimum

energy configuration

given Jij

Si= +1 Si= -1




For T



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

the free-energy valleys are identified with the pure states of the system. We can then introduce restricted averages ⟨···⟩α. Local magnetization for each state miα = ⟨σi⟩α.

Thermal average of an observable O ⟨O⟩ = ∑α wα⟨O⟩α, Distance qαβ = 1/N ∑i miα miβ.


Broken symmetries & Goldstone modes


Scalar field and broken symmetries

1-dimensional version of the Higgs potential. The x-axis represents the Higgs vev. For any value ≠0, this means that the Higgs field is on at very point in spacetime, allowing fermions to bounce off of it and hence become massive. The y-axis is the potential energy cost of the Higgs taking a particular vacuum value—we see that to minimize this energy, the Higgs wants to roll down to a non-zero vev.



Actually, because the Higgs vev can be any complex number, a more realistic picture is to plot the Higgs potential over the complex plane.



Now the minimum of the potential is a circle and the Higgs can pick any value. Higgs particles are quantum excitations—or ripples—of the Higgs field. Quantum excitations which push along this circle are called Goldstone bosons, and these represent the parts of the Higgs which are eaten by the gauge bosons.



Of course, in the Standard Model we know there are three Goldstone bosons (one each for the W+, W-, and Z), so there must be three “flat directions” in the Higgs potential. Unfortunately, I cannot fit this many dimensions into a 2D picture. The remaining Higgs particle is the excitation in the not-flat direction.



At low temperature with a scalar field there is a symmetry that is broken. There are two contributions, one in the longitudinal and one in the transverse direction. Saddle point (δQ=0) δQΔδQ+Mab,cdδQabδQcd+Tr(δQab)2+Qab(δQab)3+(δQab)4, Mab,cd derivata 2a matrice hessiana

!!" ≠ 0, ! = !! + !"!!


4-Goldstone bosons scattering

The scattering of four Goldstone bosons with zero momentum is protected by Ward identities. If the model is π2+ σ2, where ≠ 0 (propagator indicated by a wavy line) the scattering of the Goldstone bosons has zero amplitude at zero point.



This 4-point function cancels because of Ward identity and so everything is right. 1/k2 possible infrared singularities. k= 0 if the vertex is canceled, the infrared singularities are reduced. In general, if you consider the theory in 4 dimensions, e.g. a theory with propagator 1/k2 with interaction φ4 this produces IR divergences; if you consider a massless Goldstone boson this does not happen.



ddk 1k 4ddk ∫ k2 →

ddk 1k 4d 2k ∫ k2 D = 2

ddk 1k 4d 4k ∫ k2 D = 4 SAVE!

DIVERGENT!


Introduction


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.


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.


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 the symmetry group which leaves invariant the ansatz of Q (q fluctuations) break even this n replica permutation group

L

R

A


Toy model


Projected propagator

small conjugate field ε (explicit RSB)

Functional Bare propagator 1st order in ε

let us define a propagator in the subspace identified by q(x): 2 index propagator (toy model)

kinetic term


Projected propagator

x,y 0

•  Propagator induced by ε is massive (x < x1)

•  how continuous replica symmetry breaking gives rise to the diagonal p−3 singularity for small p (before x1)

•  kinetic term of order p2 on the diagonal →contribution p−2 on the diagonal of G

•  to keep the off-diagonal elements of the product of the two matrices zero, a diverging off- diagonal contribution (p−3) in G is also needed


Projected propagator with a source ε ≠ 0

At zero momentum the propagator

Propagator 4-point function

in the four point function the infrared contributions from the two diagrams with four external legs, the one from the quartic vertex and the one from two cubic vertices with a propagator flowing between, are similar but do not cancel.


Full theory


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


Ultrametric trees

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

R subspace

L-A diagonal subspace

L-A off-diag

onal subspa

ce

Volume


Topologies I can define a distance xab = the distance between a and b Generally speaking, the distance is a number → it becomes a function of 6 parameters. Simplification: given 4 indices 4 topologies (different dependences on the parameters)


Gab,ab(k,xab) •  Application of a matrix M to G •  parameterization in terms of all the variables I need; •  a set of linear equations, •  resolution and study of the solution (a little complicated). If you look at the spectrum of M, I find that the eigenvalues equal to zero and a number different from zero. Those different from zero have a point of accumulation in zero (infinite), those which are equal to zero are negative infinity, in such a way that offset.

If one looks at the simplest things, such as Gab, ab(k, xab) this is the situation: x= xmax →1/k2 0

Taxonomic structure of the tree of states (K=3)

Iterative generation of a pruned tree with K=3 RSB


Generation of a pruned tree (K=3 RSB steps)


Conclusions


So far… What next?

analysis the ultrametric structure of the

subspace where the fluctuations are of

order p−3

for finite p this propagator is

essentially given by two contributions,

their singularities canceling for p ≃ 0

This propagator, defined in the

subspace identified by q(x), turns out

to be, as expected, the projection of the

complete set of propagators in this

subspace.

toy propagator that expresses how a small

external field, explicitly breaking replica

symmetry, induces a perturbation on the

order parameter q(x)

the volume of the off-diagonal

subspace is −x one might conjecture

that, by casting the infrared behavior of

the propagators within the theory of

distributions, these singularities cancel.

Conclusions


Università degli studi di Roma “La Sapienza”

THANK YOU

The ultrametric tree of states in spin glasses: perturbative analysis and explicit generation

Andrea Lucarelli

Documents

The ultrametric tree of states in spin glasses: perturbative …€¦ · Andrea Lucarelli – The ultrametric tree of states in spin glasses H A [{g}] = ∫dxTr(A µ g(x))2 gauge