Condensed Matter and Critical Phenomena

5 Sept 2016, 09:30 → 7 Sept 2016, 18:00 Europe/Rome

Auditorium B. Touschek (INFN - Laboratori Nazionali di Frascati <!--ID_UTENTE=509-->)

Alessandro Giuliani (Univ. Roma 3), Alessandro Pizzo (Universita' di Roma "Tor Vergata"), Vieri Mastropietro (Univ. Milano)

The workshop aims at presenting recent results in the mathematical theory of critical phenomena and effective theories in condensed matter physics and statistical mechanics, including: scaling limits of discrete spin systems, percolation models and interacting random walks; effective dynamics and phase transitions in interacting quantum many body systems; disordered electrons and localization phenomena.

Invited speakers include: 
M. Aizenman, R. Bauerschmidt, D. Chelkak, G. Gallavotti, K. Gawedzki, V. Mastropietro, B. Nachtergaele, A. Pizzo,  M. Porta, B. Schlein,  A. Vichi, S. Warzel

Organizers:
A. Giuliani, V. Mastropietro, A. Pizzo
 

 

Monday, 5 September

1 Stability of Frustration-Free Ground States of Quantum Spin Systems

Abstract: We study frustration-free quantum lattice systems with a non-vanishing spectral gap above one or more (infinite-volume) ground states. The ground states are called stable if arbitrary perturbations of the Hamiltonian that are uniformly small throughout the lattice have only a perturbative effect. In the past several years such stability results have been obtained in increasing generality. We review results by Bravyi-Hastings, Bravyi-Hastings-Michalakis, and Michalakis-Zwolak, as well as recent refinements. (Joint work with Robert Sims and Amanda Young.)

Speaker: Prof. Bruno Nachtergaele (University of California, Davis)

Slides Nachtergaele.pdf

10:30 Coffee Break

2 Localization of interacting fermions with quasi-random disorder

We consider interacting electrons in a one dimensional lattice with an incommensurate Aubry-Andre' potential in the regime when the single-particle eigenstates are localized. We rigorously establish persistence of ground state localization in presence of weak many-body interaction, for almost all the chemical potentials. The proof uses a quantum many body extension of methods adopted for the stability of tori of nearly integrable hamiltonian systems, and relies on number-theoretic properties of the potential incommensurate frequency.

Speaker: Prof. Vieri Mastropietro (Universita' di Milano)

Slides Mastropietro.pdf

3 Convergence of correlations in the 2D Ising model: primary fields and the stress-energy tensor

In this talk we plan to summarize recent results on convergence of correlations functions in the critical 2D nearest-neighbor Ising model (on general planar domains) to their continuous counterparts. This includes mixed correlations of spins, disorders, fermions and energy densities (in preparation, joint with Clement Hongler (Lausanne) and Konstantin Izyurov (Helsinki)) and a discrete version of the stress-energy tensor (arXiv:1604.06339, joint with Alexander Glaznam (Tel-Aviv) and Stanislav Smirnov (Geneva)). The main technical tool is convergence theorems for discrete holomorphic spinors that are known to solve particular Riemann-type boundary value problems. In particular, one can construct all the aforementioned correlation functions and to deduce relevant CFT fusion rules starting with solutions to these Riemann-type boundary value problems in continuum.

Speaker: Prof. Dmitry Chelkak (ENS, Paris)

Slides Chelkak.pdf

13:30 Lunch

4 Precise critical exponents from Conformal bootstrap

Originally formulated in the 70's, the conformal bootstrap is the ambitious idea that one can use internal consistency conditions to carve out, and eventually solve, the space of conformal field theories. In this talk I will review recent developments in the field which have boosted this program to a new level. I will present a method to extract quantitative informations in strongly-interacting theories, such as 3D Ising, O(N) vector model and even systems without a Lagrangian formulation. I will explain how these techniques have led to the world record determination of several critical exponents.

Speaker: Prof. Alessandro Vichi ((Cern, Geneva))

16:00 Coffee Break & Discussion

Tuesday, 6 September

5 Response in the topological insulators

I shall discuss topological invariants of insulators and their response interpretation.

Speaker: Prof. Krzysztof Gawedzki (ENS de Lyon)

Slides Gawedzki.pdf

10:30 Coffee Break

6 Universality and Hall transitions in interacting fermionic systems Abstract: In this talk I will discuss the charge transport properties of weakly interacting fermionic

In this talk I will discuss the charge transport properties of weakly interacting fermionic systems, in the zero temperature and infinite volume limit. In the first part of the talk I will consider general interacting, gapped fermionic systems on periodic two-dimensional lattices.  Our theorem states that the Kubo conductivity matrix is independent of many-body interactions, provided the interaction strength is small enough. In particular, the result proves the stability of the integer quantum Hall effect against weak many-body interactions. In the second part of the talk, I will focus on the transitions between different topological Hall phases in the interacting Haldane model. The Haldane model is a graphene-like model that, in the absence of interactions, displays a non-trivial topological phase diagram. We consider the model in the presence of weak many-body interactions, and we give a rigorous construction of the renormalized transition line. Despite the nontrivial renormalization of the wave function and of the Fermi velocity, the conductivity is universal: at the renormalized critical line, both the discontinuity of the transverse conductivity and the longitudinal conductivity do not depend on the interaction. The proofs are based on a combination of cluster expansion techniques, rigorous renormalization group and Ward identities. Joint work with A. Giuliani, I. Jauslin and V. Mastropietro.

Speaker: Prof. Marcello Porta

Slides Porta.pdf

7 Invariant measures for nonlinear Schroedinger equations as limit of many body quantum states.

We prove that Gibbs measures of nonlinear Schroedinger equations of Hartree-type arise as high-temperature limit of appropriately modified thermal states in many-body quantum mechanics. In dimensions d=2,3 these Gibbs measures are supported on singular distributions and Wick ordering of the interaction is necessary. Our proof is based on a perturbative expansion in the interaction, organised in a diagrammatic representation, and on Borel resummation of the resulting series. This is a joint work with J. Froehlich, A. Knowles and V. Sohinger.

Speaker: Prof. Benjamin Schlein (Univ. Zurich)

Slides Schlein.pdf

13:30 Lunch

8 Bose Particles in a Box: Convergent Expansion of the Ground State

I shall report on some recent works where I have introduced a multi-scale analysis in the occupation numbers of particle states that provides a convergent expansion of the ground state of an interacting Bose gas in a finite box and in the mean field limiting regime. The talk will be mainly focussed on the construction of the ground state of a three-modes system.

Speaker: Prof. Alessandro Pizzo (Universita' di Roma "Tor Vergata")

Slides Pizzo.pdf

16:00 Coffee Break & Discussion

Wednesday, 7 September

9 Kondo's effect in a hierarchical s-d model and its perspectives.

The hierarchical model for the s-d model is simpler than the corresponding version of the Andrei kodel. But there is a precise relation between the RG flows for the two models which will be illustrated. Some properties of the beta function for the translation invariant (non hierarchical) Andrei's model will be also discussed.

Speaker: Prof. Giovanni Gallavotti (ROMA1)

Slides Gallavotti.pdf

10:30 Coffee Break

10 Spectral properties of random regular graphs

In this talk we plan to summarize recent results on convergence of correlations functions in the critical 2D nearest-neighbor Ising model (on general planar domains) to their continuous counterparts. This includes mixed correlations of spins, disorders, fermions and energy densities (in preparation, joint with Clement Hongler (Lausanne) and Konstantin Izyurov (Helsinki)) and a discrete version of the stress-energy tensor (arXiv:1604.06339, joint with Alexander Glaznam (Tel-Aviv) and Stanislav Smirnov (Geneva)). The main technical tool is convergence theorems for discrete holomorphic spinors that are known to solve particular Riemann-type boundary value problems. In particular, one can construct all the aforementioned correlation functions and to deduce relevant CFT fusion rules starting with solutions to these Riemann-type boundary value problems in continuum.

Speaker: Prof. Roland Bauerschmidt (University of Cambridge)

11 Order - Disorder operators in planar and almost planar graphs

Speaker: Prof. Michael Aizenman (Princeton University)

Slides Aizenman.pdf

13:30 Lunch

12 Pfaffian Correlation Functions of Planar Dimer Covers

In this talk I will explain an elementary derivation of the Pfaffian nature of boundary monomer correlation functions in the dimer-covering problem on planar graphs. The boundary monomer correlation functions are then extended into a larger family of order-disorder correlation functions which are shown to exhibit Pfaffian structure throughout the bulk. Key tools which will be discussed involve combinatorial switching symmetries which are identified through the loop-gas representation of the double dimer model, and topological implications of planarity. This is joint work with Michael Aizenman and Manuel Lainz Valcazar.

Speaker: Prof. Simone Warzel (TU Munich)

Slides Warzel.pdf

16:00 Coffee Break & Discussion