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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
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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
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Introduction
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Disordered systems: spin glasses
Biology
Proteins
Neural networks
River basins morphology
Keyword: collective behavior of a large heterogeneous system of
interacting agents
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
Networks
Finance networks
Evolution networks
Internet networks
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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).
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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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.
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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.
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…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
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Ising spin glass hamiltonian
H[{s}]Ising = -∑JijSiSj, Si=±1, i=1,…,N
Finding minimum
energy configuration
given Jij
Si= +1 Si= -1
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Ising spin glass hamiltonian
H[{s}]Ising = -∑JijSiSj, Si=±1, i=1,…,N
For T
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Ising spin glass hamiltonian
H[{s}]Ising = -∑JijSiSj, Si=±1, i=1,…,N
For T
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Ising spin glass hamiltonian
H[{s}]Ising = -∑JijSiSj, Si=±1, i=1,…,N
For T
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Ising spin glass hamiltonian
H[{s}]Ising = -∑JijSiSj, Si=±1, i=1,…,N
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.
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Broken symmetries & Goldstone modes
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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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.
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Scalar field and broken symmetries
Actually, because the Higgs vev can be any complex number, a
more realistic picture is to plot the Higgs potential over the
complex plane.
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Scalar field and broken symmetries
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.
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Scalar field and broken symmetries
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.
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Scalar field and broken symmetries
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, ! = !! + !"!!
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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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.
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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4-Goldstone bosons scattering
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.
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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4-Goldstone bosons scattering
ddk 1k 4ddk ∫ k2 →
ddk 1k 4d 2k ∫ k2 D = 2
ddk 1k 4d 4k ∫ k2 D = 4 SAVE!
DIVERGENT!
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Introduction
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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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.
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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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.
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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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
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Toy model
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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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
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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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
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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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.
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Full theory
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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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
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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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
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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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)
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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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
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Taxonomic structure of the tree of states (K=3)
Iterative generation of a pruned tree with K=3 RSB
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Generation of a pruned tree (K=3 RSB steps)
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Generation of a pruned tree (K=3 RSB steps)
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Generation of a pruned tree (K=3 RSB steps)
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Conclusions
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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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
Andrea Lucarelli – The ultrametric tree of states in spin
glasses
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Università degli studi di Roma “La Sapienza”
THANK YOU
The ultrametric tree of states in spin glasses: perturbative
analysis and explicit generation
Andrea Lucarelli