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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
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
Andrea Lucarelli – The ultrametric tree of states 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
Andrea Lucarelli – The ultrametric tree of states in spin glasses
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).
Andrea Lucarelli – The ultrametric tree of states in spin glasses
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.
…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
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
Ising spin glass hamiltonian
H[{s}]Ising = -∑JijSiSj, Si=±1, i=1,…,N
For T
Ising spin glass hamiltonian
H[{s}]Ising = -∑JijSiSj, Si=±1, i=1,…,N
For T
Ising spin glass hamiltonian
H[{s}]Ising = -∑JijSiSj, Si=±1, i=1,…,N
For T
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
Broken symmetries & Goldstone modes
Andrea Lucarelli – The ultrametric tree of states in spin glasses
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
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
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
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
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
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
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
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
Introduction
Andrea Lucarelli – The ultrametric tree of states in spin glasses
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
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
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
Toy model
Andrea Lucarelli – The ultrametric tree of states in spin glasses
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
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
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
Full theory
Andrea Lucarelli – The ultrametric tree of states in spin glasses
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
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
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
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
Andrea Lucarelli – The ultrametric tree of states in spin glasses
Generation of a pruned tree (K=3 RSB steps)
Andrea Lucarelli – The ultrametric tree of states in spin glasses
Generation of a pruned tree (K=3 RSB steps)
Andrea Lucarelli – The ultrametric tree of states in spin glasses
Generation of a pruned tree (K=3 RSB steps)
Andrea Lucarelli – The ultrametric tree of states in spin glasses
Conclusions
Andrea Lucarelli – The ultrametric tree of states in spin glasses
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
Università degli studi di Roma “La Sapienza”
THANK YOU
The ultrametric tree of states in spin glasses: perturbative analysis and explicit generation
Andrea Lucarelli