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

Transcription

1 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

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

3 Disordered systems: spin glasses Keyword: collective behavior of a large heterogeneous system of interacting agents Biology Networks Proteins Finance networks Neural networks Evolution networks River basins morphology Internet networks

4 Ising spin glass hamiltonian: symmetry and symmetry breaking H[{s}] Ising = - <i,j> J ij S i S j - h i S i, S i =±1, i=1,,n nearest neighbours quenched parameters J ij Gaussian distribution zero average variance J 2 =1/N magnetic field h the energy of a state{s i } is precisely the same as the energy of the state with every spin flipped {-s i } with h 0 the symmetry is explicitly broken: the Hamiltonian does not have the s s symmetry (Z 2 ). True beauty is a deliberate, partial breaking of symmetry (Zen proverb)

5 Ising spin glass hamiltonian: symmetry and symmetry breaking H[{s}] Ising = - <i,j> J ij S i S j, S i =±1, i=1,,n alternative definition (in the continuum) H A [{g}] = dxtr(a µ g (x)) 2 gauge group G Z 2 gauge field A µ (x) J gauge transform g(x) σ We obtain the previous definition when the gauge group G is Z 2, we are on the lattice and we consider the strong coupling limit. In many cases Gribov ambiguity tells us that H A (g) has many minima, therefore H J (σ ) has an exponentially large number of minima.

6 Ising spin glass hamiltonian H[{s}] Ising = - <i,j> J ij S i S, j Si=±1, i=1,,n Finding minimum energy configuration given J ij S i = +1 S i = -1

7 Ising spin glass hamiltonian H[{s}] Ising = - <i,j> J ij S i S, j Si=±1, i=1,,n For T<T c a very complex spin- glass phase appears, characterized by the existence of infinitely many relevant equilibrium states Local magnetization for each state m i α = σ i α states unrelated to one another by simple symmetries and separated by very high free-energy barriers. Distance q αβ = 1/N i m i α m i β free energy valleys system pure states

8 Scalar field and broken symmetries 1-D version of the Higgs potential Higgs potential over the complex plane Higgs particles are quantum excitations (ripples) of the Higgs field. Quantum excitations which push along the circle are called Goldstone bosons. 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 <Q ab > 0, Q=Q+ Q

9 Correlation functions and divergencies When a continuous symmetry is spontaneously broken, new massless scalar particles appear in the spectrum of possible excitations (Goldstone theorem). 4 Goldstone bosons scattering For D=4 theory, e.g. a theory with interaction φ 4 : 1/k 2 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?

10 The replica approach H[{s}] Ising = - <i,j> J ij S i S j - h i S i, S i =±1, i=1,,n 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. Free energy density in powers of Q Functional in terms of q [0,1] Stationarity equations wr to q for T<T c Replica symmetry breaking (RSB) is related to the probability of measuring a given value q for the overlap < s i > a < s i > b between two states a and b differing by a finite amount in free energy.

11 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

12 Longitudinal, anomalous, replicon the fluctuations of the order parameter Q around the RSB saddle point are usually divided into three families L invariant under the permutations symmetry of the n replicas Order parameter Q Fluctuations around the RSB saddle point Anomalous A R break even this n replica permutation group

13 Projected propagator an explicit RSB can be introduced in the theory by adding to the effective free energy the term Functional kinetic term small conjugate field ε (explicit RSB) Bare propagator G(x,y;p) x=y<x 1, p=0 distribution x,y<x 1, p 0 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.

14 Green functions and divergencies Propagator (mass matrix with diag kinetic term) -1 Replicon GF Longitudinal Anomalous GF O(p -3 ) divergences u -1 ultrametric prefactor Propagator 2 order O(p -2 ) divergences x -2 ultrametric prefactor Disconnected diagram 2 order u= min(x,y) v= max(x,y)

15 Ultrametric trees 4 ultrametric indices different possibilities of arranging them on an ultrametric tree Different topologies x ab = distance between a and b G studied in all phase space (# terms ~ 10 6 ) 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)

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

17 Diagrammatic expansion

18 Bethe mean field theory vs mean field theory Diagrammatical expansion from Bethe MFT and not from the naïve MFT. Bethe mean field Mean field approx 1. Random quantum Hamiltonian 2. T=0 Spin Glass in a magnetic field 3. Random magnetic Ising ferromagnets at T=0 Esperimenti ad alta the transition energia from localized to extended [frontiera states di alta for energia] a random quantum Hamiltonian. the transition Esperimenti from di alta the glassy phase precisione to the paramagnetic (a bassa energia) phase in zero temperature spin glasses [frontiera in a magnetic di alta field. intensità] in the study of random magnetic Esperimenti Ising ferromagnets di alta at zero precisione temperature (a bassa there is a region energia) where many extensively [frontiera different alta local intensità] minima do exist Localized states do not exist in the usual mean field approximation In mean field approximation at T=0 we are always in the glassy phase This does not exist in the standard mean field theory approach.

19 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

20 Bethe approximation Let s consider point i Let s call V i the set of the points k such that A i,k =1 (the neighbourhood of i). Let s remove the spin i Let s suppose that in this situation the spin in V i are uncorrelated (they must be correlated when the spin i is present) cavity magnetizations m C (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

21 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.

22 M-layered model Infinite range M-layered model Short range M-layered model NxM spins s i a i=1,,n a=1,,m NxM spins s i a i=1,,n a=1,,m The Hamiltonian is The Hamiltonian is where for each edge i,k we have a quenched permutation p a (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.

23 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/M 2 two loops M=1 standard lattice, usual D-dimensional lattice: all genera 1/M = loop expansion 1/M k =genus k expansion

24 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

25 Linear and Loop (Bethe) correlations at T=0 h h Linear Bethe computation at T=0 S 1 S 2 S 3 è S 1 S 3 L 2 S 1 S 2 S 3 S 4 L 1 L 4 L 3 è 1 loop Bethe computation at T=0 S 1 S 4

26 Bethe approx in a d-dimensional lattice 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. 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.

27 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.

28 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 T ab,cd =Q ab Q cd

29 Eigenvalues and projectors on eigenvalues L Anomalous A R (multiplicity n-1) (multiplicity n(n-3)/2) We are interested in the correlations at distance k, so we want to compute T k : in the limit n 0 λ 1 =λ 2 (so the first addend can be rewritten)

30 Connected and disconnected functions Zero temperature ferromagnes in a random magnetic field Ising spins SG in field RFIM

31 two spins Hamiltonian Disconnected function At T=0 we have the ground state σ 1*, σ 2 * of the Hamiltonian. We can group the result of minimization in three different scenarios. magnetizations (mean value)

32 two spins Hamiltonian Disconnected function Assumiamo che c, n 1, n 2, k 1, k 2 ~O(1) siano quantità finite. Ansatz per la funzione di correlazione

33 Conclusions and perspectives Expansion around Bethe solution finite z RSB What next So far To do: Ferromagnetic random magnetic systems at T=0 comparison between analytical results and simulations Spin glass random magnetic systems at T=0 the one dimensional study is the same of the previous case Localization starting level (1/N correction to the spectral density of the Lagrangian on RRG) SG at T=0 Localization Jamming Expansion around Bethe Jamming first steps

34 Università degli studi di Roma La Sapienza The ultrametric tree of states and computation of correlation functions in spin glasses Andrea Lucarelli