Table Of ContentUniversita` degli Studi Roma Tre
7
1
0
2
DOTTORATO DI RICERCA IN MATEMATICA
n
a
J
9 XXIX CICLO
1
]
h
p
-
h
t
a
m
[
Interacting fermions on the half-line:
1
v boundary counterterms and boundary corrections
6
5
3
5
0
.
1
0
7
1
Candidate: Giovanni Antinucci
:
v
i
X Supervisor: Prof. Alessandro Giuliani
r
a
Coordinator: Prof. Luigi Chierchia
DEC, 2016
3
La spensieratezza
va stroncata alla nascita.
- -
ANTONIOREZZA-FrattoX
4
Acknowledgements
The work behind this thesis lasted three years, one of which I spent abroad: so let me take the
opportunity to thank first of all the Universit`a degli Studi di Roma Tre, that gave me the pos-
sibility of attending different educational experiences. Besides, I wish to express my gratitude to
those who hosted me: the Universit´e de Lyon 1, where I spent three months (October - December
2015), supported by the A*MIDEX project Hypathie (n. ANR-11-IDEX-0001-02) funded by the
”Investissements d’Avenir” 25 French Government program, managed by the French National Re-
searchAgency(ANR),havingtheopportunitytoenjoytheverystimulatingendexcitingclimateof
the Probability Group of Professor Toninelli; and the Universit´e de Geneve, and SwissMap project,
that funded me to attend, as a visiting student, the wonderful Masterclass in Statistical Physics
2015-2016, from January to June 2016.
Il mio piu` grande e sincero ringraziamento va al Prof. A. Giuliani, per la fiducia che ha avuto
in me proponendomi questo problema, e per l’interesse, la disponibilit`a, e l’infinita pazienza con
cui mi ha seguito: senza il suo supporto, scientifico e umano, la realizzazione di questo lavoro non
sarebbe stata possibile.
Ringrazio il Prof. V. Mastropietro, per le stimolanti discussioni e per il cruciale suggerimento
di inserire un non-local boundary counterterm.
Ringrazio il Prof. M. Porta, per tutto l’entusiasmo che mi ha trasmesso durante le mie, seppur
brevi, visite a Zurigo.
Ringrazio il Dott. I. Jauslin, che ha reso meno traumatico e piu` veloce il mio inserimento nel
mondo della Fisica Matematica e del RG, grazie alle lunghissime e coinvolgenti discussioni sempre
affrontate con interesse e passione.
Ringrazio il Prof. Gallavotti: `e grazie a lui, ai suoi consigli e al suo esempio che ho intrapreso la
strada della Fisica Matematica e, in particolare, del Gruppo di Rinormalizzazione.
Infine,ringrazioiProfessoriM.Correggi,G.Dell’Antonio,G.Panati,A.Tetaperavermisempre
datolapossibilit`adipartecipareallemolteconferenzedaloroorganizzate,eperaversempreaccolto
con grande interesse qualsiasi mia domanda o semplice curiosit`a.
5
6
Abstract
Recentyearswitnessedanextensivedevelopmentofthetheoryofthecriticalpointintwo-dimensional
statistical systems, which allowed to prove existence and conformal invariance of the scaling limit
for two-dimensional Ising model and dimers in planar graphs. Unfortunately, we are still far from
a full understanding of the subject: so far, exact solutions at the lattice level, in particular de-
terminant structure and exact discrete holomorphicity, play a cucial role in the rigorous control
of the scaling limit. The few results about not-integrable (interacting) systems at criticality are
still unable to deal with finite domains and boundary corrections, which are of course crucial for
getting informations about conformal covariance. In this thesis, we address the question of adapt-
ing constructive Renormalization Group methods to non-integrable critical systems in d = 1+1
dimensions. We study a system of interacting spinless fermions on a one-dimensional semi-infinite
lattice, which can be considered as a prototype of the Luttinger universality class with Dirichlet
Boundary Conditions. We develop a convergent renormalized expression for the thermodynamic
observables in the presence of a quadratic boundary defect counterterm, polynomially localized at
the boundary. In particular, we get explicit bounds on the boundary corrections to the specific
ground state energy.
7
8
Contents
1 Introduction 11
1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 The model and the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 The outline of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Interacting fermions on the line 23
2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.1 Definition and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.2 Free Hamiltonian diagonalization and free propagator . . . . . . . . . . . . . 25
2.2 Perturbation theory and Grassmann integral formulation. . . . . . . . . . . . . . . . 26
2.2.1 Perturbation theory and Trotter’s formula . . . . . . . . . . . . . . . . . . . . 26
2.2.2 How to compute truncated expectations . . . . . . . . . . . . . . . . . . . . . 31
2.2.3 The determinant expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Interacting case: the multiscale analysis . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 Ultraviolet and infrared regimes, effective potential . . . . . . . . . . . . . . . 35
2.3.2 Quasi-particles and multiscale expansion . . . . . . . . . . . . . . . . . . . . . 37
2.3.3 Gallavotti-Nicol`o trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.4 Non-renormalizedexpansion,non-perturbativeestimatesandclassificationof
the divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.1 Localization and Renormalization operator . . . . . . . . . . . . . . . . . . . 48
2.4.2 Scale h integration and dressed theory . . . . . . . . . . . . . . . . . . . . . . 53
2.4.3 The renormalized tree expansion and renormalized bounds . . . . . . . . . . 57
2.4.4 Flow of running coupling constants . . . . . . . . . . . . . . . . . . . . . . . . 63
3 Interacting Fermions on the half line 69
3.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1.1 Definition and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1.2 Free Hamiltonian diagonalization and free propagator . . . . . . . . . . . . . 70
3.2 Interacting case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2.1 Trotter’s formula and Grassmann integration . . . . . . . . . . . . . . . . . . 74
3.2.2 Multiscale decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2.3 Properties of single-scale free propagators . . . . . . . . . . . . . . . . . . . . 82
3.3 Non-renormalized expansion and properties of kernels . . . . . . . . . . . . . . . . . 83
3.3.1 Properties of the kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
9
10 CONTENTS
3.4 Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.1 ”Preliminar” localization: L . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
B
3.4.2 Definition of localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.4.3 Scale h integration and dressed theory . . . . . . . . . . . . . . . . . . . . . . 93
3.4.4 The renormalized tree expansion . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.4.5 Renormalized bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.4.6 Proof of the main theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.4.7 Flow of running coupling constants and functions . . . . . . . . . . . . . . . . 119
4 Conclusion 127
4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A Estimates of single scale propagators 131
B Finite volume localization (PBC) 135
C Gram representation 141
D Breaking of DBC: a counter-example 145
E Finite volume localization definition, DBC 147
F Weighted integration and dimensional gain 151
