PHYSICAL REVIEW E 88, 022139 (2013)
Critical behavior of the ideal-gas Bose-Einstein condensation in the Apollonian network
I. N. de Oliveira,
1
T. B. dos Santos,
1
F. A. B. F. de Moura,
1
M. L. Lyra,
1,2
and M. Serva
3,4
1
Instituto de F´ ısica, Universidade Federal de Alagoas, 57072-970 Macei´ o, AL, Brazil
2
Laboratoire de Physique de la Mati` ere Condens´ ee, UMR CNRS 7643, Ecole Polytechnique, 91128 Palaiseau, Cedex, France
3
Departamento de Biof´ ısica e Farmacologia, Universidade Federal do Rio Grande do Norte, 59072-970, Natal-RN, Brazil
4
Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Universit` a dell’Aquila, 67010 L’Aquila, Italy
(Received 5 February 2013; published 26 August 2013)
We show that the ideal Boson gas displays a finite-temperature Bose-Einstein condensation transition in
the complex Apollonian network exhibiting scale-free, small-world, and hierarchical properties. The single-
particle tight-binding Hamiltonian with properly rescaled hopping amplitudes has a fractal-like energy spectrum.
The energy spectrum is analytically demonstrated to be generated by a nonlinear mapping transformation. A
finite-size scaling analysis over several orders of magnitudes of network sizes is shown to provide precise
estimates for the exponents characterizing the condensed fraction, correlation size, and specific heat. The critical
exponents, as well as the power-law behavior of the density of states at the bottom of the band, are similar to
those of the ideal Boson gas in lattices with spectral dimension d
s
= 2ln(3)/ln(9/5) 3.74.
DOI: 10.1103/PhysRevE.88.022139 PACS number(s): 05.30.Jp, 67.85.Jk, 64.60.aq, 64.60.F−
I. INTRODUCTION
Bose-Einstein condensation (BEC) is one of the most
remarkable quantum phenomena on which a macroscopic
fraction of the bosonic particles constituting a physical system
occupies a single quantum state, thus leading to the emergence
of macroscopic spontaneous coherence. The production of
gaseous BEC of cold weakly interacting atoms in a magnetic
trap [1,2] represented a landmark in the physics history
corroborating that the BEC is a purely quantum phenomenon
that can take place even when interparticle interactions are
negligible. Nowadays, BEC has also been reported in solid-
state quasiparticles systems such as excitons, antiferro, and
ferromagnetic magnons [3–6], which has stimulated additional
studies concerning the universal features in the vicinity of the
BEC transition.
The scaling behavior characterizing the Bose-Einstein
condensation of an ideal gas has been a longstanding
issue addressed by several authors in the framework of
phase transitions and critical phenomena [7–12]. It has been
demonstrated that there is a precise correspondence between
the asymptotic properties of the thermodynamic quantities
in the vicinity of the transition temperature and those of
the spherical model of ferromagnetism [7]. Considering a
single-particle density of states (DOS) having a power-law
behavior DOS ∝ E
σ
at the band bottom, the exponents
characterizing the singular behavior of several quantities
have been obtained [7,8], with σ = d/2 − 1 for particles
enclosed in a d -dimensional box. One remarkable result is
that the condensed fraction vanishes linearly as the reduced
temperature t = (T
c
− T )/T
c
→ 0 irrespective to the value of
σ , where T
c
is the transition temperature below which a finite
fraction of the particles condensate at the ground state. On the
other hand, the correlation length diverges as ξ ∝ t
−ν
, with
ν = 1/2σ for 2 <d< 4 and ν = 1/2 for d> 4. The specific
heat exponent is finite at the transition. For d< 4, the specific
heat is continuous and a negative exponent α =−(1 − σ )/σ
characterizes its cusp singularity, where C
v
(T ) − C
v
(T
c
) ∝
|t |
−α
. For d> 4, it develops a jump discontinuity with α =
−(σ − 1). At d = 4, a logarithmic singularity sets up in the
specific heat. These exponents are modified by the presence
of interparticle interactions. In particular, the condensed
fraction decreases sublinearly, as reported in superfluid helium
experiments [9].
In spite of the well-established critical behavior of the ideal
gas BEC transition in homogeneous lattices, the corresponding
scenario in complex inhomogeneous lattices is still under-
explored. Within this context, exact analytical expressions for
the thermodynamic properties of the ideal gas on the star and
wheel networks have been recently reported [13]. The presence
of a gap between the ground and excited states is responsible
for the emergence of a low-temperature condensed phase, a
feature also shared by networks composed of interconnected
linear chains [14–18]. In the star and wheel networks, the
critical behavior is mean-field-like. The condensed fraction
vanishes linearly when approaching the transition, the specific
heat is discontinuous, and the condensed fraction at the
transition temperature scales with the number of lattice sites
as N
−1/2
.
BEC in scale-free networks are much less understood.
These complex networks having a power-law distribution of
site connectivity represent an important class of lattice models,
which has contributed to the understanding of transport and
information flow within systems of many degrees of freedom
[19–23]. In this context, the deterministic Apollonian network
has attracted much attention due to its scale-free and small-
world properties [24–28]. The thermodynamic properties of
the ideal electron gas on the Apollonian network reflects the
complex structure of the single-particle DOS, such as the
presence of δ-like singularities, gaps, and mini-bands [29,30].
On the other hand, when considering the same hopping
amplitude between any pair of connected sites, the ideal
boson gas was shown to present only a condensed phase
in the thermodynamic limit, with no finite-temperature BEC
transition [31]. This feature is related to the divergence of
the ground-state energy with the network size due to the
presence of sites with a diverging number of connections in the
thermodynamic limit. Therefore, the actual critical behavior of
the BEC transition of the ideal gas in scale-free networks is
still an open issue.
022139-1 1539-3755/2013/88(2)/022139(7) ©2013 American Physical Society