Ann. Henri Poincar´e 18 (2017), 2873–2903
2017 The Author(s).
This article is an open access publication
published online May 18, 2017
Annales Henri Poincar´e
Complex Bosonic Many-Body Models:
Overview of the Small Field Parabolic Flow
Tadeusz Balaban, Joel Feldman, Horst Kn¨orrer and
Abstract. This paper is a contribution to a program to see symmetry
breaking in a weakly interacting many boson system on a three-dimen-
sional lattice at low temperature. It provides an overview of the analysis,
given in Balaban et al. (The small ﬁeld parabolic ﬂow for bosonic many-
body models: part 1—main results and algebra, arXiv:1609.01745, 2016,
The small ﬁeld parabolic ﬂow for bosonic many-body models: part 2—
ﬂuctuation integral and renormalization, arXiv:1609.01746, 2016), of the
‘small ﬁeld’ approximation to the ‘parabolic ﬂow’ which exhibits the for-
mation of a ‘Mexican hat’ potential well.
It is our long-term goal to rigorously demonstrate symmetry breaking in a gas
of bosons hopping on a three-dimensional lattice. Technically, to show that the
correlation functions decay at a nonintegrable rate when the chemical poten-
tial is suﬃciently positive, the nonintegrability reﬂecting the presence of a long
range Goldstone boson mediating the interaction between quasiparticles in the
superﬂuid condensate. It is already known [19,20] that the correlation func-
tions are exponentially decreasing when the chemical potential is suﬃciently
negative. See, for example,  and [30, §19] for an introduction to symme-
try breaking in general, and [1,18,23,28] as general references to Bose–Einstein
condensation. See [17,21,26,29] for other mathematically rigorous work on the
We start with a brief, formula free, summary of the program and its
current state. Then we’ll provide a more precise, but still simpliﬁed, discussion
of the portion of the program that controls the small ﬁeld parabolic ﬂow.
Research supported in part by the Natural Sciences and Engineering Research Council of
Canada and the Forschungsinstitut f¨ur Mathematik, ETH Z¨urich.