The temporal ultraviolet limit

Abstract. This paper is a contribution to a program to see symmetry breaking in a weakly interacting many boson system on a three-dimensional lattice at low temperature. It provides an overview of the analysis, given in Balaban et al. (The small field parabolic flow for bosonic manybody 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 formation 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 potential 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 functions are exponentially decreasing when the chemical potential is sufficiently negative. See, for example, [22] and [30, §19] for an introduction to symmetry 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 subject.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.

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“…Our main result is: If ǫ > 0 and v 0 are small enough and L is large enough, there exists a 7 µ * = O(v 0 ) , such that for all 8…”

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confidence: 96%

Complex Bosonic Many-Body Models: Overview of the Small Field Parabolic Flow

Bałaban

Feldman

Knörrer

et al. 2017

Ann. Henri Poincaré

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“…In fact, the lowest order contribution to Ŵ (h) 1,1;J ∆ (0, 0) is given by the sum of the diagrams in Fig. 23, with at least one propagator at scale h. After the same considerations made for the calculation of the beta function for µ h (see Section C.3) we find that, at lowest order, 4 .…”

Section: A Ultraviolet Flow In the Condensate Phasementioning

confidence: 60%

“…bounds"), but the possible Borel summability of the series remains an outstanding open problem. A program addressing this issue has been started by T. Balaban and collaborators (see [4] and references therein), but the solution is still far from being reached: the idea is to apply the Wilsonian RG in the form developed by Balaban in a series of works dedicated to the construction of theories with a broken continuous symmetry [1,2,3], but the new technical problems arising in the case of a complex gaussian measure (as the one appearing in the Bose gas) seem to be serious obstacles to the solution.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Renormalization Theory of a Two Dimensional Bose Gas: Quantum Critical Point and Quasi-Condensed State

Cenatiempo

Giuliani

2014

J Stat Phys

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We present a renormalization group construction of a weakly interacting Bose gas at zero temperature in the two-dimensional continuum, both in the quantum critical regime and in the presence of a condensate fraction. The construction is performed within a rigorous renormalization group scheme, borrowed from the methods of constructive field theory, which allows us to derive explicit bounds on all the orders of renormalized perturbation theory. Our scheme allows us to construct the theory of the quantum critical point completely, both in the ultraviolet and in the infrared regimes, thus extending previous heuristic approaches to this phase. For the condensate phase, we solve completely the ultraviolet problem and we investigate in detail the infrared region, up to length scales of the order (λ 3 ρ 0 ) −1/2 (here λ is the interaction strength and ρ 0 the condensate density), which is the largest length scale at which the problem is perturbative in nature. We exhibit violations to the formal Ward Identities, due to the momentum cutoff used to regularize the theory, which suggest that previous proposals about the existence of a non-perturbative non-trivial fixed point for the infrared flow should be reconsidered.

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“…is an S invariant, particle-number preserving function with real valued kernels and with E 0 (0, 0) = 0, that 8 We show, in Lemma D.2, that, to leading order, µ * is θ times (1.5). 9 That is,R…”

Section: The Starting Point Setupmentioning

confidence: 78%

The Small Field Parabolic Flow for Bosonic Many-body Models: Part 1—Main Results and Algebra

Bałaban

Feldman

Knörrer

et al. 2018

Ann. Henri Poincaré

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This paper is a contribution to a program to see symmetry breaking in a weakly interacting many Boson system on a three dimensional lattice at low temperature. It is part of an analysis of the "small field" approximation to the "parabolic flow" which exhibits the formation of a "Mexican hat" potential well. Here we state the main result of this analysis, outline the strategy of the proof, which uses a renormalization group flow, and perform the first, algebraic, part of a renormalization group step.

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The temporal ultraviolet limit

Cited by 4 publications

References 43 publications

Complex Bosonic Many-Body Models: Overview of the Small Field Parabolic Flow

Complex Bosonic Many-Body Models: Overview of the Small Field Parabolic Flow

Renormalization Theory of a Two Dimensional Bose Gas: Quantum Critical Point and Quasi-Condensed State

The Small Field Parabolic Flow for Bosonic Many-body Models: Part 1—Main Results and Algebra

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