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
€
= 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 T-1/3-) 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{ }) = −12
siθ iJijθ js j − Hisii
∑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(Hi + θ iJijθ jm j
0
j∑ )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
€
Z = e−βH si{ }( )
si{ }
∑
Stationary phase approximationThe free energy is given in the stationary phase
approximation by
The inverse susceptibility matrix is given by
€
Γ mi{ }( ) = −12
miθ iJijθ jm j +1
2β(1 − mi)ln(1 − mi) + (1+ mi)ln(1+ mi)[ ] +
i∑
ij∑
12zβ
lndet δ ij − βJijθ iθ j (1 − m j2)[ ]
€
ij−1 =
∂Γ mi{ }( )∂mi∂m j
Results of the stationary phase approximation
€
Λ−1
T − Tc
=1 −1z
dλρ (λ ) f (λ )λ2
(T − λ )(TC − λ )∫
€
Tc = Λ −1z
dλ f (λ )λ
1−λΛ
∫
€
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 Λ 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
ρ)
Λc
Distribution of the spectral gap
For networks with
the spectral gap is non-self-averaging but its distribution is stable.
€
P(θ ) ∝ θ −γ SF
J(r r i,
r r j ) = e−|
r r i −
r r j | / d 0
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[ ]
€
Nδ S −1 →∞
thenΨ →∞
S. Bradde F. Caccioli L. Dall’Asta G. Bianconi PRL 2010
Random Transverse Ising model
€
H = −J2
aijσ izσ j
z − ε iσ ix − hσ i
z
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 ρe) 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 i
z
€
mθ ,εz = σ i
zθ 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
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 i
zσ αz
€
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∑ = J
Bα ,i
Bα ,i2 +εα
2tanhβ
α∈N( i)\ j∑ Bα ,i
2 +εα2
€
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
€
1L
logΞ = 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 <3 the critical coupling Jc(T=0) decreases as the cutoff ξ 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 less or equal to 3
G. Bianconi JSTAT 2012
Enhancement of Tc with the increasing value of the exponential cutoff
The critical temperature for 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 =U2
ni(ni −1) − μni
⎡ ⎣ ⎢
⎤ ⎦ ⎥− t
i∑ τ ij ai
ij∑ 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 =U2
ni(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 =U2
ni(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
€
Hi = Hi(0) + θ i γ t (ai + ai
+)
E i(0)(n) = E (0)(n)
E (0)(n*) =0 if μ < 0
−μ n *+12
Un * (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
θn *
U(n * −1) − μ+
n * +1μ −Un *
⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
tc = Uθθ 2
μ /U − n *[ ] (n * −1) − μ /U[ ]μ /U +1
with 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 =U2
ni(ni −1) − μni − t τ ij (ai + ai+)ψ j
j∑
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥+ 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 → ∞asN →∞
Mean-field phase diagram of random scale-free network
=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 <3.
• 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
€
> 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 <3.
• 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