space and imaginary time. Such a formulation would be the right one in order to deal with additional, possibly non-translationally invariant, perturbations, including weak random potentials (chosen in such a way that the non-interacting spectral gap is *not* closed by the randomness).

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The work of A. G. and of V. M. has been supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ERC CoG UniCoSM, grant agreement n.724939). V. M. acknowledges support also from the Gruppo Nazionale di Fisica Matematica (GNFM). The work of M. P. has been supported by the Swiss National Science Foundation, via the grant "Mathematical Aspects of Many-Body Quantum Systems".

[^1]: Communicated by Michael Aizenman.


INTRODUCTION
============

TropGeneDB is an information system initially developed by CIRAD (French Agricultural Research Centre for International Development) to manage various kinds of data on tropical crops ([@gks1105-B1]).

TropGeneDB can record crop information on:molecular markers, quantitative trait loci (QTLs), genetic and physical mapsgenetic diversity studiesphenotypic diversity studies based on agro-morphological traits or traits measuring susceptibility/resistance to aggression (diseases, salinity, drought)geographic origin, parentage, collection

The TropGeneDB project was initiated using the ACEDB database management system ([@gks1105-B2]), but we migrated to MySQL to follow the evolution of computer technology. TropGeneDB is organized on a crop basis. Since the first version of TropGeneDB ([@gks1105-B1]) containing three running modules (cocoa, banana and sugarcane), six additional modules have been implemented: coconut, coffee, cotton, oil palm, rice and rubber tree. A sorghum module exists, but it is still in private access.

New crop-specific Web consultation interfaces have been designed to allow q