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

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Abstract

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 with the critical one, that is the amount of the condensate is negligible with respect to the amount of the whole particles. All those states should satisfy the KMS boundary condition with respect to the natural dynamics generated by the formal Pure Hopping Hamiltonian. The existence of such a dynamics, which is a delicate issue, is provided in detail.

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Notes

  1. Concerning the appearance of the BEC for the cases under consideration, for further details we refer the reader to Proposition 3.1, and the related analysis in Sect. 3.

  2. By passing to a subsequence, it is easily shown that the sequence of finite volume PF eigenvectors as above, converges by compactness to a PF weight, see Proposition 4.1 of [8]. In all the situations considered in the present paper, the sequence of finite volume PF eigenvectors for the chosen exhaustion converges point-wise to a PF weight without passing to subsequences, as we will see below.

  3. The homogeneous cases \({\mathbb Z}^d\) is extensively treated in literature, whereas the recurrent examples for which the PF weight is normalisable (i.e. a PF eigenvector, necessarily unique up a phase factor) are trivial.

  4. It is possible to show that the PF weight is not unique, even for the Adjacency of a Comb Graph in the transient case.

  5. The choice of entire functions is only for the sake of convenence, see the last part of the proof of Theorem 4.5.

  6. It is also customary to fix the activity \(z:=e^{\beta \mu }\), instead of the chemical potential.

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Acknowledgments

The author kindly acknowledges Yuri Safarov for help in proving Proposition 5.7. He is also grateful to two anonymous referees whose suggestions contributed to improve the presentation of the present paper.

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  1. Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133, Roma, Italy

    Francesco Fidaleo

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  1. Francesco Fidaleo
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Correspondence to Francesco Fidaleo.

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Fidaleo, F. Harmonic Analysis on Inhomogeneous Amenable Networks and the Bose–Einstein Condensation. J Stat Phys 160, 715–759 (2015). https://doi.org/10.1007/s10955-015-1263-4

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