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Vieri Mastropietro

Publications and source records attributed to Vieri Mastropietro.

At least 73 records · Page 4Linked to original sources
arXiv2010

Drude weight in non solvable quantum spin chains

For a quantum spin chain or 1D fermionic system, we prove that the Drude weight D verifies the universal Luttinger liquid relation $v_s^2=D/κ$, where $κ$ is the susceptibility and $v_s$ is the Fermi velocity. This result is proved by rigorous Renormalization Group methods and is true for any weakly interacting system, regardless its integrablity. This paper, combined with a previous paper of the same authors, completes the proof of the Luttinger liquid conjecture for such systems.

cond-mat.stat-mech↗
arXiv2010

Anomalous behavior in an effective model of graphene with Coulomb interactions

We analyze by exact Renormalization Group (RG) methods the infrared properties of an effective model of graphene, in which two-dimensional massless Dirac fermions propagating with a velocity smaller than the speed of light interact with a three-dimensional quantum electromagnetic field. The fermionic correlation functions are written as series in the running coupling constants, with finite coefficients that admit explicit bounds at all orders. The implementation of Ward Identities in the RG scheme implies that the effective charges tend to a line of fixed points. At small momenta, the quasi-particle weight tends to zero and the effective Fermi velocity tends to a finite value. These limits are approached with a power law behavior characterized by non-universal critical exponents.

cond-mat.str-el↗
arXiv2010

Rigorous construction of ground state correlations in graphene: renormalization of the velocities and Ward Identities

We consider the 2D Hubbard model on the honeycomb lattice, as a model for single layer graphene with screened Coulomb interactions; at half filling and weak coupling, we construct its ground state correlations by a convergent multiscale expansion, rigorously excluding the presence of magnetic or superconducting instabilities or the formation of a mass gap. The Fermi velocity, which can be written in terms of a convergent series expansion, remains close to its non-interacting value and turns out to be isotropic. On the contrary, the interaction produces an asymmetry between the two components of the charge velocity, in contrast with the predictions based on relativistic or continuum approximations.

cond-mat.str-el↗
arXiv2010

Lattice gauge theory model for graphene

The effects of the electromagnetic (e.m.) electron-electron interactions in half-filled graphene are investigated in terms of a lattice gauge theory model. By using exact Renormalization Group methods and lattice Ward Identities, we show that the e.m. interactions amplify the responses to the excitonic pairings associated to a Kekulé distortion and to a charge density wave. The effect of the electronic repulsion on the Peierls-Kekulé instability, usually neglected, is evaluated by deriving an exact non-BCS gap equation, from which we find evidence that strong e.m. interactions among electrons facilitate the spontaneous distortion of the lattice and the opening of a gap.

cond-mat.str-el↗
arXiv2009

The 2D Hubbard model on the honeycomb lattice

We consider the 2D Hubbard model on the honeycomb lattice, as a model for a single layer graphene sheet in the presence of screened Coulomb interactions. At half filling and weak enough coupling, we compute the free energy, the ground state energy and we construct the correlation functions up to zero temperature in terms of convergent series; analiticity is proved by making use of constructive fermionic renormalization group methods. We show that the interaction produces a modification of the Fermi velocity and of the wave function renormalization without changing the asymptotic infrared properties of the model with respect to the unperturbed non-interacting case; this rules out the possibility of superconducting or magnetic instabilities in the ground state. We also prove that the correlations verify a Ward Identity similar to the one for massless Dirac fermions, up to asymptotically negligible corrections and a renormalization of the charge velocity.

math-ph↗
arXiv2008

Luttinger liquid fixed point for a 2D flat Fermi surface

We consider a system of 2D interacting fermions with a flat Fermi surface. The apparent conflict between Luttinger and non Luttinger liquid behavior found through different approximations is resolved by showing the existence of a line of non trivial fixed points, for the RG flow, corresponding to Luttinger liquid behavior; the presence of marginally relevant operators can cause flow away from the fixed point. The analysis is non-perturbative and based on the implementation, at each RG iteration, of Ward Identities obtained from local phase transformations depending on the Fermi surface side, implying the partial vanishing of the Beta function.

cond-mat.str-el↗
arXiv2006

Non perturbative Adler-Bardeen Theorem

The Adler-Bardeen theorem has been proved only as a statement valid at all orders in perturbation theory, without any control on the convergence of the series. In this paper we prove a nonperturbative version of the Adler-Bardeen theorem in $d=2$ by using recently developed technical tools in the theory of Grassmann integration.

hep-th↗
arXiv2005

Gap generation in the BCS model with finite range temporal interaction

In the [BCS] paper the theory of superconductivity was developed for the BCS model, in which the (instantaneous) interaction is only between fermions of opposite momentum and spin. Such model was analyzed by variational methods, finding that a superconducting behavior is energetically favorable. Subsequently it was claimed that in the thermodynamic limit the BCS model is equivalent to the (exactly solvable) quadratic mean field BCS model; a rigorous proof of this claim is however still lacking. In this paper we consider the BCS model with a finite range temporal interaction, and we prove rigorously its equivalence with the mean field BCS model in the thermodinamic limit if the range is long enough, by a (uniformly convergent) perturbation expansion about mean field theory.

cond-mat.str-el↗
arXiv2005

Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities

We prove the existence of small amplitude periodic solutions, for a large Lebesgue measure set of frequencies, in the nonlinear beam equation with a weak quadratic and velocity dependent nonlinearity and with Dirichlet boundary conditions. Such nonlinear PDE can be regarded as a simple model describing oscillations of flexible structures like suspension bridges in presence of an uniform wind flow. The periodic solutions are explicitly constructed by means of a perturbative expansion which can be considered the analogue of the Lindstedt series expansion for the invariant tori in classical mechanics. The periodic solutions are not analytic but defined only in a Cantor set, and resummation techniques of divergent powers series are used in order to control the small divisors problem.

math.FA↗
arXiv2005

Rigorous proof of Luttinger liquid behavior in the 1d Hubbard model

We give the first rigorous (non perturbative) proof of Luttinger liquid behavior in the one dimensional Hubbard model, for small repulsive interaction and values of the density different from half filling. The analysis is based on the combination of multiscale analysis with Ward identities bases on a hidden and approximate local chiral gauge invariance. No use is done of exact solutions or special integrability properties of the Hubbard model, and the results can be in fact easily generalized to include non local interactions, magnetic fields or interaction with external potentials

cond-mat.str-el↗
arXiv2004

Ward identities and Chiral anomaly in the Luttinger liquid

Systems of interacting non relativistic fermions in $d=1$, as well as spin chains or interacting bidimensional Ising models, verify an hidden approximate Gauge invariance which can be used to derive suitable Ward identities. Despite the presence of corrections and anomalies, such Ward identities can be implemented in a Renormalization Group approach and used to exploit nontrivial cancellations which allow to control the flow of the running coupling constants; in particular this is achieved combining Ward identities, Dyson equations and suitable correction identities for the extra terms appearing in the Ward identities, due to the presence of cutoffs breaking the local gauge symmetry. The correlations can be computed and show a Luttinger liquid behavior characterized by non universal critical indices, so that the general Luttinger liquid construction for one dimensional systems is completed without any use of exact solutions. The ultraviolet cutoff can be removed and a Quantum Field Theory corresponding to the Thirring model is also constructed.

cond-mat.other↗
arXiv2004

Periodic solutions for completely resonant nonlinear wave equations

We consider the nonlinear string equation with Dirichlet boundary conditions $u_{xx}-u_{tt}=ϕ(u)$, with $ϕ(u)=Φu^{3} + O(u^{5})$ odd and analytic, $Φ\neq0$, and we construct small amplitude periodic solutions with frequency $ø$ for a large Lebesgue measure set of $ø$ close to 1. This extends previous results where only a zero-measure set of frequencies could be treated (the ones for which no small divisors appear). The proof is based on combining the Lyapunov-Schmidt decomposition, which leads to two separate sets of equations dealing with the resonant and nonresonant Fourier components, respectively the Q and the P equations, with resummation techniques of divergent powers series, allowing us to control the small divisors problem. The main difficulty with respect the nonlinear wave equations $u_{xx}-u_{tt}+ M u = ϕ(u)$, $M\neq0$, is that not only the P equation but also the Q equation is infinite-dimensional

math.DS↗
arXiv2002

Non-universality in Ising models with four spin interaction

We consider two bidimensional classical Ising models, coupled by a weak interaction bilinear in the energy densities of the two systems; the model contains, as limiting cases, the Ashkin-Teller and the Eight-vertex models for certain values of their parameters. We write the energy-energy correlations and the specific heat as Grassman integrals formally describing Dirac 1+1 dimensional interacting massive fermions on a lattice, and an expansion based on Renormalization Group is written for them, convergent up to temperatures very close to the critical temperature for small coupling. The asymptotic behaviour is determined by critical indices which are continuous functions of the coupling.

cond-mat.stat-mech↗
arXiv2001

Peierls instability with electron-electron interaction: the commensurate case

We consider a quantum many-body model describing a system of electrons interacting with themselves and hopping from one ion to another of a one dimensional lattice. We show that the ground state energy of such system, as a functional of the ionic configurations, has local minima in correspondence of configurations described by smooth ${π\over p_F}$ periodic functions, if the interaction is repulsive and large enough and $p_F$ is the Fermi momentum of the electrons. This means physically that a $d=1$ metal develop a periodic distortion of its reticular structure (Peierls instability). The minima are found solving the Eulero-Lagrange equations of the energy by a contraction method.

cond-mat.str-el↗
arXiv2001

Incommensurate Charge Density Waves in the adiabatic Hubbard-Holstein model

The adiabatic, Holstein-Hubbard model describes electrons on a chain with step $a$ interacting with themselves (with coupling $U$) and with a classical phonon field $\f_x$ (with coupling $ł$). There is Peierls instability if the electronic ground state energy $F(\f)$ as a functional of $\f_x$ has a minimum which corresponds to a periodic function with period ${π\over p_F}$, where $p_F$ is the Fermi momentum. We consider ${p_F\overπa}$ irrational so that the CDW is {\it incommensurate} with the chain. We prove in a rigorous way in the spinless case, when $ł,U$ are small and ${U\overł}$ large, that a)when the electronic interaction is attractive $U<0$ there is no Peierls instability b)when the interaction is repulsive $U>0$ there is Peierls instability in the sense that our convergent expansion for $F(\f)$, truncated at the second order, has a minimum which corresponds to an analytical and ${π\over p_F}$ periodic $\f_x$. Such a minimum is found solving an infinite set of coupled self-consistent equations, one for each of the infinite Fourier modes of $\f_x$.

cond-mat.str-el↗
arXiv1999

Anomalous BCS equation for a Luttinger superconductor

In the context of the Anderson theory of high T_c cuprates, we develop a BCS theory for Luttinger liquids. If the Luttinger interaction is much stronger than the BCS potential we find that the BCS equation is quite modified compared to usual BCS equation for Fermi liquids. In particular T_c predicted by the BCS equation for Luttinger liquids is quite higher than the usual T_c for Fermi liquids.

cond-mat.str-el↗