In this paper we construct entire solutions uε to the Cahn–Hilliard equation −ε2Δ(−ε2Δu+W′(u))+W″(u)(−ε2Δu+W′(u))=ε4λε(1−uε), under the volume constraint ∫R3(1−uε)2dx=82π2cε, with cε→1 as ε→0, whose nodal set approaches the Clifford Torus, that is the Torus with radii of ratio 1/2 embedded in R3, as ε→0. It is crucial that the Clifford Torus is a Willmore hypersurface and it is non-degenerate, up to conformal transformations. The proof is based on the Lyapunov–Schmidt reduction and on careful geometric expansions of the Laplacian.

%B Journal of Differential Equations %V 262 %P 5306 - 5362 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022039617300530 %R https://doi.org/10.1016/j.jde.2017.01.026 %0 Journal Article %J arXiv preprint arXiv:1601.07093 %D 2016 %T Critical points of a perturbed Otha-Kawasaki functional %A Matteo Rizzi %B arXiv preprint arXiv:1601.07093 %G eng %0 Thesis %D 2016 %T Qualitative properties and construction of solutions to some semilinear elliptic PDEs %A Matteo Rizzi %K moving planes method, maximum principle, Lyapunov-Schmidt reduction, Willmore surfaces, Otha-Kawasaki functional %X This thesis is devoted to the study of elliptic equations. On the one hand, we study some qualitative properties, such as symmetry of solutions, on the other hand we explicitly construct some solutions vanishing near some fixed manifold. The main techniques are the moving planes method, in order to investigate the qualitative properties and the Lyapunov-Schmidt reduction. %I SISSA %G en %1 35500 %# MAT/05 %$ Submitted by mrizzi@sissa.it (mrizzi@sissa.it) on 2016-07-11T13:48:21Z No. of bitstreams: 1 tesi_finale.pdf: 1142117 bytes, checksum: 481f2b0713509231cd3a6f8ead92bf59 (MD5) %0 Journal Article %J Annali della Scuola Normale Superiore di Pisa. Classe di scienze %D 2016 %T Symmetry properties of some solutions to some semilinear elliptic equations %A Farina, Alberto %A Andrea Malchiodi %A Matteo Rizzi %B Annali della Scuola Normale Superiore di Pisa. Classe di scienze %I Classe di Scienze %V 16 %P 1209–1234 %G eng %0 Journal Article %J Phys. Rev. A 81 (2010) 062335 %D 2010 %T Homogeneous binary trees as ground states of quantum critical Hamiltonians %A Pietro Silvi %A Vittorio Giovannetti %A Simone Montangero %A Matteo Rizzi %A J. Ignacio Cirac %A Rosario Fazio %XMany-body states whose wave-function admits a representation in terms of a uniform binary-tree tensor decomposition are shown to obey to power-law two-body correlations functions. Any such state can be associated with the ground state of a translational invariant Hamiltonian which, depending on the dimension of the systems sites, involve at most couplings between third-neighboring sites. A detailed analysis of their spectra shows that they admit an exponentially large ground space.

%B Phys. Rev. A 81 (2010) 062335 %I American Physical Society %G en_US %U http://hdl.handle.net/1963/3909 %1 800 %2 Physics %3 Condensed Matter Theory %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-07-27T07:32:21Z\\nNo. of bitstreams: 1\\n0912.0466v1.pdf: 336415 bytes, checksum: 7220d67cfb58f794aa50037140db23e6 (MD5) %R 10.1103/PhysRevA.81.062335 %0 Journal Article %J New J. Phys. 12 (2010) 075018 %D 2010 %T Homogeneous multiscale entanglement renormalization ansatz tensor networks for quantum critical systems %A Matteo Rizzi %A Simone Montangero %A Pietro Silvi %A Vittorio Giovannetti %A Rosario Fazio %XIn this paper, we review the properties of homogeneous multiscale entanglement renormalization ansatz (MERA) to describe quantum critical systems.We discuss in more detail our results for one-dimensional (1D) systems (the Ising and Heisenberg models) and present new data for the 2D Ising model. Together with the results for the critical exponents, we provide a detailed description of the numerical algorithm and a discussion of new optimization\\nstrategies. The relation between the critical properties of the system and the tensor structure of the MERA is expressed using the formalism of quantum channels, which we review and extend.

%B New J. Phys. 12 (2010) 075018 %I IOP Publishing %G en_US %U http://hdl.handle.net/1963/4067 %1 335 %2 Physics %3 Condensed Matter Theory %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-09-16T09:50:44Z\\nNo. of bitstreams: 1\\n10081611392421027.pdf: 1233170 bytes, checksum: 39da921aa8c098e7c8d6451b0ce08c15 (MD5) %R 10.1088/1367-2630/12/7/075018 %0 Journal Article %J Phys. Rev. B 77 (2008) 245105 %D 2008 %T Fulde-Ferrell-Larkin-Ovchinnikov pairing in one-dimensional optical lattices %A Matteo Rizzi %A Marco Polini %A Miguel A. Cazalilla %A M.R. Bakhtiari %A Mario P. Tosi %A Rosario Fazio %XSpin-polarized attractive Fermi gases in one-dimensional (1D) optical lattices are expected to be remarkably good candidates for the observation of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. We model these systems with an attractive Hubbard model with population imbalance. By means of the density-matrix renormalization-group method we compute the pairing correlations as well as the static spin and charge structure factors in the whole range from weak to strong coupling. We demonstrate that pairing correlations exhibit quasi-long range order and oscillations at the wave number expected from FFLO theory. However, we also show by numerically computing the mixed spin-charge static structure factor that charge and spin degrees of freedom appear to be coupled already for small imbalance. We discuss the consequences of this coupling for the observation of the FFLO phase, as well as for the stabilization of the quasi-long range order into long-range order by coupling many identical 1D systems, as in quasi-1D optical lattices.

%B Phys. Rev. B 77 (2008) 245105 %G en_US %U http://hdl.handle.net/1963/2694 %1 1406 %2 Physics %3 Condensed Matter Theory %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-07-14T09:46:14Z\\nNo. of bitstreams: 1\\n0712.3364v1.pdf: 305409 bytes, checksum: 13081c375909264afc4c0282b9de9f68 (MD5) %R 10.1103/PhysRevB.77.245105 %0 Journal Article %J Phys. Rev. Lett. 98 (2007) 030404 %D 2007 %T Luther-Emery Phase and Atomic-Density Waves in a Trapped Fermion Gas %A Gao Xianlong %A Matteo Rizzi %A Marco Polini %A Rosario Fazio %A Mario P. Tosi %A Vivaldo L. Jr. Campo %A Klaus Capelle %XThe Luther-Emery liquid is a state of matter that is predicted to occur in one-dimensional systems of interacting fermions and is characterized by a gapless charge spectrum and a gapped spin spectrum. In this Letter we discuss a realization of the Luther-Emery phase in a trapped cold-atom gas. We study by means of the density-matrix renormalization-group technique a two-component atomic Fermi gas with attractive interactions subject to parabolic trapping inside an optical lattice. We demonstrate how this system exhibits compound phases characterized by the coexistence of spin pairing and atomic-density waves. A smooth crossover occurs with increasing magnitude of the atom-atom attraction to a state in which tightly bound spin-singlet dimers occupy the center of the trap. The existence of atomic-density waves could be detected in the elastic contribution to the light-scattering diffraction pattern.

%B Phys. Rev. Lett. 98 (2007) 030404 %G en_US %U http://hdl.handle.net/1963/2056 %1 2140 %2 Physics %3 Condensed Matter Theory %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-09-06T11:46:28Z\\nNo. of bitstreams: 1\\ncond-mat0609346v1.pdf: 218755 bytes, checksum: 06a409d540e05ece03bbac85198ee19c (MD5) %R 10.1103/PhysRevLett.98.030404 %0 Journal Article %J Phys. Rev. B 73 (2006) 100502(R) %D 2006 %T 4e-condensation in a fully frustrated Josephson junction diamond chain %A Matteo Rizzi %A Vittorio Cataudella %A Rosario Fazio %XFully frustrated one-dimensional diamond Josephson chains have been shown [B. Dou\\\\c{c}ot and J. Vidal, Phys. Rev. Lett. {\\\\bf 88}, 227005 (2002)] to posses a remarkable property: The superfluid phase occurs through the condensation of pairs of Cooper pairs. By means of Monte Carlo simulations we analyze quantitatively the Insulator to $4e$-Superfluid transition. We determine the location of the critical point and discuss the behaviour of the phase-phase correlators. For comparison we also present the case of a diamond chain at zero and 1/3 frustration where the standard $2e$-condensation is observed.

%B Phys. Rev. B 73 (2006) 100502(R) %G en_US %U http://hdl.handle.net/1963/2400 %1 2297 %2 Physics %3 Condensed Matter Theory %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-11-13T08:27:31Z\\nNo. of bitstreams: 1\\n0511423v1.pdf: 197406 bytes, checksum: 754c3576bbb6a549f4dac12eeb8fc92d (MD5) %R 10.1103/PhysRevB.73.100502