Bose–einstein Condensation on Inhomogeneous Amenable Graphs

Fidaleo, Francesco; Guido, Daniele; Isola, Tommaso

doi:10.1142/s0219025711004389

Cited by 16 publications

(75 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 8.4 of [8]), whereas Proposition 6.3 is compatible with the fact that the Adjacency A N is transient. The latter property is the necessary and sufficient condition for the existence of locally normal states exhibiting BEC, see Theorem 4.5.…”

Section: Proposition 63 Let λ N ≥ a S N Such That Limsupporting

confidence: 58%

“…Here, the first limit exists by Lemma 3.4 of [8] and is 0 by normalisation. The second one is meaningful directly by the definition of the IDS.…”

Section: The Dynamics Generated By the Bogoliubov Transformationsmentioning

confidence: 89%

“…Concerning the infinite volume limit of the finite volume density, we get (cf. [3,7,8]) in the condensation regime for each sequence of the chemical potential μ n → 0,…”

Section: Theorem 33 For An Hamiltonian H ≥ 0 On a Graph G With 0 ∈ σmentioning

confidence: 99%

“…Fix any PF weight v for A G which exists by compactness (cf. Section 4 of [8]). Consider the continuous function…”

Section: Proposition 43 If a G Is Recurrent Then For Each X ∈ Gmentioning

confidence: 99%

“…Thus, the above considerations are nothing but the naive explanation of the fact that, after cooling below the critical temperature, the system undergoes a "dimensional transition" (proven in details below) governed by the possible difference between the geometrical and PF dimensions, naturally appearing in inhomogeneous cases. Some of the results listed above are proved in [8] for a class of amenable comb graphs. The investigation of the BEC for free Bosons on homogeneous Cayley Trees has been started in the seminal paper [19].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Harmonic Analysis on Inhomogeneous Amenable Networks and the Bose–Einstein Condensation

Fidaleo

2015

J Stat Phys

Self Cite

View full text Add to dashboard Cite

We study in detail relevant spectral properties of the adjacency matrix of inhomogeneous amenable networks, and in particular those arising by negligible additive perturbations of periodic lattices. The obtained results are deeply connected to the systematic investigation of the Bose-Einstein condensation for the so called Pure Hopping model describing the thermodynamics of Cooper pairs in arrays of Josephson junctions. After a careful investigation of the infinite volume limits of the finite volume adjacency matrix corresponding to the (opposite of the) Hamiltonian of the system, the main results can be summarised as follows. First, the appearance of the Hidden Spectrum for the Integrated Density of the States in the region close to the bottom of the Hamiltonian, implies that the critical density is always finite. Second, we show that the Bose-Einstein condensation can appear if and only if the adjacency matrix is transient, and not just when the critical density is finite. We can then exhibit examples of networks for which condensation effects can appear in a natural way, even if the critical density is infinite and vice-versa, that is when the critical density is finite but the system does not admit any locally normal state exhibiting condensation. Contrarily to the known homogeneous examples, we also exhibit networks whose geometrical dimension is less than 3, for which the condensation takes place. Due to inhomogeneity, the spatial distribution of the condensate described by the shape of the ground state wave-function (i.e. the Perron-Frobenius weight), is non homogeneous as well: the particles condensate even in the configuration space. Such a spatial distribution of the condensate is connected with the Perron-Frobenius dimension defined in a natural way. For systems for which the critical density is finite and the adjacency matrix is transient, we show that, if the Perron-Frobenius dimension is greater that the geometrical one, we can have condensation only if the mean density of the state is infinite. Conversely, in the opposite situation when the geometrical dimension exceeds the Perron-Frobenius one, the condensation appears only for states with mean density coinciding A Berta con profonda stima e gratitudine. B Francesco Fidaleo

show abstract

Section: Proposition 63 Let λ N ≥ a S N Such That Limsupporting

confidence: 58%

“…Here, the first limit exists by Lemma 3.4 of [8] and is 0 by normalisation. The second one is meaningful directly by the definition of the IDS.…”

Section: The Dynamics Generated By the Bogoliubov Transformationsmentioning

confidence: 89%

“…Concerning the infinite volume limit of the finite volume density, we get (cf. [3,7,8]) in the condensation regime for each sequence of the chemical potential μ n → 0,…”

Section: Theorem 33 For An Hamiltonian H ≥ 0 On a Graph G With 0 ∈ σmentioning

confidence: 99%

“…Fix any PF weight v for A G which exists by compactness (cf. Section 4 of [8]). Consider the continuous function…”

Section: Proposition 43 If a G Is Recurrent Then For Each X ∈ Gmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Harmonic Analysis on Inhomogeneous Amenable Networks and the Bose–Einstein Condensation

Fidaleo

2015

J Stat Phys

Self Cite

View full text Add to dashboard Cite

show abstract

A C∗-algebra of geometric operators on self-similar CW-complexes. Novikov–Shubin and L2-Betti numbers

Cipriani¹,

Guido²,

Isola³

2009

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

A class of CW-complexes, called self-similar complexes, is introduced, together with C * -algebras A j of operators, endowed with a finite trace, acting on square-summable cellular j -chains. Since the Laplacian j belongs to A j , L 2 -Betti numbers and Novikov-Shubin numbers are defined for such complexes in terms of the trace. In particular a relation involving the Euler-Poincaré characteristic is proved. L 2 -Betti and Novikov-Shubin numbers are computed for some self-similar complexes arising from self-similar fractals.

show abstract

Harmonic analysis on perturbed Cayley Trees

Fidaleo

2011

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

We study some spectral properties of the adjacency operator of non-homogeneous networks. The graphs under investigation are obtained by adding density zero perturbations to the homogeneous Cayley Trees. Apart from the natural mathematical meaning, such spectral properties are relevant for the Bose Einstein Condensation for the pure hopping model describing arrays of Josephson junctions on non-homogeneous networks. The resulting topological model is described by a one particle Hamiltonian which is, up to an additive constant, the opposite of the adjacency operator on the graph. It is known that the Bose Einstein condensation already occurs for unperturbed homogeneous Cayley Trees. However, the particles condensate on the perturbed graph, even in the configuration space due to non-homogeneity. Even if the graphs under consideration are exponentially growing, we show that it is enough to perturb in a negligible way the original graph in order to obtain a new network whose mathematical and physical properties dramatically change. Among the results proved in the present paper, we mention the following ones. The appearance of the Hidden Spectrum near the zero of the Hamiltonian, or equivalently below the norm of the adjacency. The latter is related to the value of the critical density and then with the appearance of the condensation phenomena. The investigation of the recurrence/transience character of the adjacency, which is connected to the possibility to construct locally normal states exhibiting the Bose Einstein condensation. Finally, the study of the volume growth of the wave function of the ground state of the Hamiltonian, which is nothing but the generalized Perron Frobenius eigenvector of the adjacency. This Perron Frobenius weight describes the spatial distribution of the condensate and its shape is connected with the possibility to construct locally normal states exhibiting the Bose Einstein condensation at a fixed density greater than the critical one.

show abstract

Bose–einstein Condensation on Inhomogeneous Amenable Graphs

Cited by 16 publications

References 11 publications

Harmonic Analysis on Inhomogeneous Amenable Networks and the Bose–Einstein Condensation

Harmonic Analysis on Inhomogeneous Amenable Networks and the Bose–Einstein Condensation

A C∗-algebra of geometric operators on self-similar CW-complexes. Novikov–Shubin and L2-Betti numbers

Harmonic analysis on perturbed Cayley Trees

Contact Info

Product

Resources

About