Spectral properties of complex networks and classical/quantum phase transitions

Ginestra Bianconi Department of Physics, Northeastern University, Boston

ETC Trento WorkshopSpectral properties of complex networks

Trento 23-29 July, 2012

Complex topologies affect the behavior of critical phenomena

Scale-free degree distribution change the critical behavior of the

Ising model, Percolation, epidemic spreading on annealed networks

Spectral propertieschange the synchronization properties,

epidemic spreading on quenched networks

Nishikawa et al.PRL 2003

Outline of the talk

• Generalization of the Ginsburg criterion for spatial complex networks (classical Ising model)

• Random Transverse Ising model on annealed and quenched networks

• The Bose-Hubbard model on annealed and quenched networks

How do critical phenomenaon complex networks change if we include spatial interactions?

Annealed uncorrelatedcomplex networks

In annealed uncorrelated complex networks, we assign to each node an expected degree

Each link is present with probability pij

The degree ki a node i is a Poisson variable with mean i

€

pij =θ iθ j

θ N

€

θ = k

θ 2 = k(k −1)

Boguna, Pastor-Satorras PRE 2003

Ising model in annealedcomplex networks

The Ising model on annealed complex networks has Hamiltonian given by

The critical temperature is given by

The magnetization is non-homogeneous€

Tc = Jθ 2

θ= J

k(k −1)

k

€

si = tanh β θ iJS + hi( )[ ]

€

H = −J

2 θ Nsiθ iθ js j − hisi

i

∑i≠ j

∑

G. Bianconi 2002,S.N. Dorogovtsev et al. 2002, Leone et al. 2002, Goltsev et al. 2003,Lee et al. 2009

Critical exponents of the Ising model on

complex topologies

M C(T<Tc)

>5 |Tc-T|1/2 Jump at Tc |Tc-T|-1

=5 |T-Tc|1/2/(|ln|TcT||)1/2 1/ln|Tc-T| |Tc-T|-1

3<<5 |Tc-T|1/(-1) |Tc-T|-)/(-3) |Tc-T|-1

=3 e-2T/<> T2e-4T/<> T-1

2<<3 T3-) T-1)/(3-) T-1

But the critical fluctuations still remain mean-field !

€

P(k) ∝ k −γ

Goltsev et al. 2003

Ensembles of spatial complex networks

The function J(d) can be measured in real spatial

networks

€

pij =θ iθ jJ(

r r i,

r r j )

1+θ iθ jJ(r r i,

r r j )

≅θ iθ jJ(r r i,

r r j )

The maximally entropic network with spatial structure has link probability given by

Airport Network Bianconi et al. PNAS 2009

J(d)

Annealead Ising model in spatial complex networks

The linking probability of spatial complex networks is chosen to be

The Ising model on spatial annealed complex networks has Hamiltonian given by

We want to study the critical fluctuations in this model as a function of the typical range of the interactions

€

pij = θ iθ jJ(r r i,

r r j )

€

H( si{ }) = −1

2siθ iJijθ js j − H isi

i

∑i≠ j

∑

Stability of the mean-field approximation

The partition function is given by

The magnetization in the mean field approximation is given by

The susceptibility is then evaluated by stationary phase approximation €

mi0 = tanh β(H i + θ iJijθ jm j

0

j

∑ ) ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

Z = e−βH si{ }( )

si{ }

∑

Stationary phase approximation

The free energy is given in the stationary phase approximation by

The inverse susceptibility matrix is given by

€

Γ mi{ }( ) = −1

2miθ iJijθ jm j +

1

2β(1 − mi)ln(1− mi) + (1+ mi)ln(1+ mi)[ ] +

i

∑ij

∑

1

2zβlndet δ ij − βJijθ iθ j (1 − m j

2)[ ]

€

χij−1 =

∂Γ mi{ }( )

∂mi∂m j

Results of the stationary phase approximation

€

χΛ−1

T − Tc

=1 −1

zdλ

ρ (λ ) f (λ )λ2

(T − λ )(TC − λ )∫

€

Tc = Λ −1

zdλ f (λ )

λ

1−λ

Λ

∫

€

f (λ ) = N uiλ ui

λ uiΛui

Λ

i

∑

We project the results into the base of eigenvalues and eigenvectors u of the matrix pij.The critical temperature Tc is given by

where is the maximal eigenvalue of the matrix pij and

The inverse susceptibility is given by

Critical fluctuations

We assume that the spectrum is given by

is the spectral gap and c the spectral edge.

Anomalous critical fluctuations sets in only if the gap vanish in the thermodynamic limit, and S<1

For regular lattice S =(d-2)/2 S<1 only if d<4

The effective dimension of complex networks is deff =2S +2

€

ρ(λ ) ∝ (λ c − λ )δ S

Δ = Λ − λ c

c

Distribution of the spectral gap

For networks with

the spectral gap is non-self-averaging but its distribution is stable.

€

P(θ ) ∝ θ −γSF

J(r r i,

r r j ) = e−|

r r i −

r r j | / d0

SF=4,d0=1 SF=6 d0=1

Criteria for onset anomalouscritical fluctuations

In order to predict anomalous critical fluctuations we introduce the quantity

If

and anomalous fluctuations sets in.

€

Ψ= limN → ∞

χ −1Λ

T − Tceff = lim

N → ∞1 − ΔN

δ S −1 C2 − C1[ ]

€

ΔNδ S −1 →∞

then

Ψ →∞

S. Bradde F. Caccioli L. Dall’Asta G. Bianconi PRL 2010

Random Transverse Ising model

€

H = −J

2aijσ i

zσ jz − ε iσ i

x − hσ iz

i

∑i

∑ij

∑

• This Hamiltonian mimics the Superconductor-Insulator phase transition in a granular superconductor

(L. B. Ioffe, M. Mezard PRL 2010,M. V. Feigel’man, L. B. Ioffe, and M. Mézard PRE 2010)

• To mimic the randomness of the onsite noise • we draw ei from a ( ) r e distribution.

• The superconducting phase transition would correspond with the phase with spontaneous magnetization

in the z direction.

Scale-free structural organization of oxygen interstitials in La2CuO4+ y

Fratini et al. Nature 2010

16K Tc=16K

RTIM on an Annealed complex network

€

aij ⏐ → ⏐ pij =θ iθ j

θ N

€

S =θ i

θ Ni

∑ σ iz

€

mθ ,εz = σ i

z

θ i =θ ,ε i =ε=

JSθ + h

(JSθ + h)2 +ε 2tanh(β (JSθ + h)2 +ε 2 )

In the annealed network model we can substitute in the Hamiltonian

The order parameter is

The magnetization depends on the expected degree q

G. Bianconi, PRE 2012

The critical temperature

€

1 = Jθ 2

θdε ρ(ε )∫ tanh(βε )

ε

€

if γ < 3θ 2

θ≈ ξ 3−γ → ∞

then Tc ∝ Jθ 2

θ= Jξ 3−γ → ∞

€

p(θ ) ∝ θ −γe−θ /ξ

Equation for Tc

Complex network topology

Scaling of Tc

€

ξ ∝| g − gc |−ν

G. Bianconi, PRE 2012

Solution of the RTIM on quenched network

€

H i, jcavity = −ε iσ i

x − εασ αx + Bασ α

z

α∈N (i)\ j

∑ + Jσ izσ α

z

€

H i, jcavity −MF = −ε iσ i

x − Jσ ix σ α

z

α∈N ( i)\ j

∑

Bij = J σ αz

α∈N ( i)\ j

∑ = JBα ,i

Bα ,i2 +εα

2tanhβ

α∈N ( i)\ j

∑ Bα ,i2 +εα

2

€

B0

B= J

tanhβεα

εα

=α∈P

∏P

∑ Ξ

On the critical line if we apply an infinitesimal field at the periphery of the network, the cavity field at a given site is given by

€

1

LlogΞ = 0

Dependence of the phase diagram from the cutoff of the degree distribution

For a random scale-free network

In general there is a phase transition at zero temperature.

Neverthelessfor l<3 the critical coupling Jc(T=0) decreases as the cutoff x increases.

€

P(k) ∝ k −λ e−k /ξ

The system at low temperature is in a Griffith Phase described by a replica-symmetry brokenPhase in the mapping to the random polymer problem

The replica-symmetry broken

phasedecreases in size with increasing

values of the cutoff for

power-law exponent g less or

equal to 3 G. Bianconi JSTAT 2012

Enhancement of Tc with the increasing value of the exponential cutoff

The critical temperature for l less or equal to 3 Increases with increasing exponential cutoffof the degree distribution

€

1 = Jk(k −1)

kdε ρ (ε )∫ tanh(βε )

ε

€

if λ < 3k(k −1)

k≈ ξ 3−λ →∞

then Tc ∝ Jk(k −1)

k= Jξ 3−λ →∞

Bose-Hubbard model on complex networks

€

ˆ H =U

2ni(ni −1) − μni

⎡ ⎣ ⎢

⎤ ⎦ ⎥− t

i

∑ τ ij aiij

∑ a j+

U on site repulsion of the Bosons,m chemical potential t coefficient of hoppingtij adjacency matrix of the network

Optical lattices

Optical lattice are nowadays use to localize cold atoms That can hop between sites by quantum tunelling.

These optical lattices have been use to test the behavior of quantum models such as the Bose-Hubbard model which was first realized with cold atoms by Greiner et al. in 2002.

The possible realization of more complex network topologies to localize cold atoms remains an open problem. Here we want to show the consequences on the phase diagram of quantum phase transition defined on complex networks.

Bose-Hubbard model: a challenge

Absorption images of multiple matter wave interface pattern as a function of the depth of the potential of the optical lattice

Experimental evidence

Theoretical approaches

The solution of the Bose-Hubbard model even on a Bethe latticeRepresent a challenge, available techniques are mean-field, dynamical mean-field model, quantum cavity model

Greiner,Mandel,Esslinger, Hansh, Bloch Nature 2002

Semerjian, Tarzia, Zamponi PRE 2009

Mean field approximation

with

on annealed network

€

aia j+ ≈ ai a j

+ + ai a j+ − ai a j

+

≈ aiψ j + a j+ψ i −ψ i ψ j

€

ai = ai+ =ψ i

€

ˆ H MF −A =U

2ni(ni −1) − μni

⎡ ⎣ ⎢

⎤ ⎦ ⎥− t

i

∑ pij (aiψ j + a j+ψ i −ψ iψ j )

ij

∑

€

pij =θ iθ j

θ N

Mean-field Hamiltonian and order parameter on a annealed network

€

ˆ H = H ii

∑ + θ Nt γ 2

H i =U

2ni(ni −1) − μni − tθ i γ (ai + ai

+)

γ =1

θ Nθ iψ i

i

∑ Order parameter of the phase transition

Perturbative solution of the effective single site Hamiltonian

€

H i = H i(0) + θ i γ t (ai + ai

+)

E i(0)(n) = E (0)(n)

E (0)(n*) =0 if μ < 0

−μ n *+1

2Un * (n * −1) if μ ∈(U(n * −1),Un*)

⎧ ⎨ ⎪

⎩ ⎪

E i(2) = γ 2 θ i

2 t 2 n *

U(n * −1) − μ+

n *+1

μ −Un *

⎛

⎝ ⎜

⎞

⎠ ⎟

Mean-field solution of the B-H model on annealed complex network

€

E = E (0)(n*) + m2 γ 2

m2

θ Nt=1+ t

θ 2

θ

n *

U(n * −1) − μ+

n * +1

μ −Un *

⎛

⎝ ⎜

⎞

⎠ ⎟

€

tc = Uθ

θ 2

μ /U − n *[ ] (n * −1) − μ /U[ ]

μ /U +1with μ /U ∈[n * −1,n*]

The critical line is determined by the line in which the mass term goes to zero m (tc,U,m)=0

There is no Mott-Insulator phase as long as the second Moment of the expected degree distribution diverges

Mean-field solution on quenched network

€

H MF =U

2ni(ni −1) − μni − t τ ij (ai + ai

+)ψ jj

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥+ τ ijψ iψ j

i, j

∑i

∑

€

ψ i = ai =t

UF(μ,U) τ ijψ j

j

∑

F(μ,U) =μ +U

[μ − n *U][U(n * −1) − μ]with μ ∈[U(n * −1),Un*]

€

tc

UF(μ,U)Λ =1

Mott phaset

UF(μ,U)Λ <1

Critical lines and phase diagram

Maximal Eigenvalue of the adjacency matrix on networks

• Random networks

• Apollonian networks€

Λ ∝ kmax →const regular random networks

∞ random Poisson graphs, scale − free networks

⎧ ⎨ ⎩

€

Λ → ∞as

N →∞

Mean-field phase diagram of random scale-free network

l=2.2N=100

N=1,000

N=10,000

Halu, Ferretti, Vezzani, Bianconi EPL 2012

Bose-Hubbard model on Apollonian network

The effective Mott-Insulator phase decreases with network size and disappear in the thermodynamic limit

References

• S. Bradde, F. Caccioli, L. Dall’Asta and G. BianconiCritical fluctuations in spatial networksPhys. Rev. Lett. 104, 218701 (2010).

• A. Halu, L. Ferretti, A. Vezzani G. Bianconi Phase diagram of the Bose-Hubbard Model on Complex Networks EPL 99 1 18001 (2012)

• G. Bianconi Supercondutor-Insulator Transition on Annealed Complex Networks Phys. Rev. E 85, 061113 (2012).

• G. Bianconi Enhancement of Tc in the Superconductor-Insulator Phase Transition on Scale-Free Networks JSTAT 2012 (in press) arXiv:1204.6282

Conclusions

• Critical phase transitions when defined on complex networks display new phase diagrams

• The spectral properties and the degree distribution play a crucial role in determining the phase diagram of critical phenomena in networks

• We can generalize the Ginsburg criterion to complex networks• The Random Transverse Ising Model (RTIM) on scale-free networks

with exponential cutoff has a critical temperature that depends on the cutoff if the power-law exponent <3g .

• The Bose-Hubbard model on quenched network has a phase diagram that depend on the spectral properties of the network

• This open new perspective in studying the interplay between spectral properties and classical/ quantum phase transition in networks

Lattices and quasicrystal

A lattice is a regular pattern of points and links repeating periodically in finite dimensions

Scale-free networks

€

λ > 3€

P(k) ∝ k −λ

with

€

1 < λ < 2

with

with

€

2 < λ < 3€

k finite

k 2 finite

€

k finite

k 2 →∞

€

k →∞

k 2 →∞

Conclusions

• Critical phase transitions when defined on complex networks display new phase diagrams

• The Random Transverse Ising Model (RTIM) on scale-free networks with exponential cutoff has a critical temperature that depends on the cutoff if the power-law exponent <3l .

• We have characterized the Bose-Hubbard model on annealed and quenched networks by the mean-field model

• This open new perspective in studying other quantum phase transitions such as rotor models, quantum spin-glass models on complex networks

• Experimental implementation of potentials describing complex networks could open new scenario for the realization of cold atoms multi-body states with new phase diagrams