W. de Siqueira Pedra

Bounds on the discrete spectrum of lattice Schrödinger operators

We discuss the validity of the Weyl asymptotics—in the sense of two-sided bounds—for the size of the discrete spectrum of (discrete) Schrodinger operators on the d-dimensional, d ≥ 1, cubic lattice Zd at large couplings. We show that the Weyl asymptotics can be violated in any spatial dimension d ≥ 1—even if the semi-classical number of bound states is finite. Furthermore, we prove for all dimensions d ≥ 1 that, for potentials well behaved at infinity and fulfilling suitable decay conditions, the Weyl asymptotics always hold. These decay conditions are mild in the case d ≥ 3 while stronger for d = 1, 2. It is well known that the semi-classical number of bound states is—up to a constant—always an upper bound on the size of the discrete spectrum of Schrodinger operators if d ≥ 3. We show here how to construct general upper bounds on the number of bound states of Schrodinger operators on Zd from semi-classical quantities in all space dimensions d ≥ 1 and independently of the positivity-improving property of ...

Microscopic conductivity of lattice fermions at equilibrium. I. Non-interacting particles

We consider free lattice fermions subjected to a static bounded potential and a time- and space-dependent electric field. For any bounded convex region ℛ ⊂ ℝd (d ≥ 1) of space, electric fields E within R drive currents. At leading order, uniformly with respect to the volume R of R and the particular choice of the static potential, the dependency on E of the current is linear and described by a conductivity (tempered, operator-valued) distribution. Because of the positivity of the heat production, the real part of its Fourier transform is a positive measure, named here (microscopic) conductivity measure of R, in accordance with Ohm’s law in Fourier space. This finite measure is the Fourier transform of a time-correlation function of current fluctuations, i.e., the conductivity distribution satisfies Green–Kubo relations. We additionally show that this measure can also be seen as the boundary value of the Laplace–Fourier transform of a so-called quantum current viscosity. The real and imaginary parts of con...

A microscopic two-band model for the electron-hole asymmetry in high-Tc superconductors and reentering behavior

To our knowledge there is no rigorously analyzed microscopic model explaining the electron-hole asymmetry of the critical temperature seen in high-Tc cuprate superconductors – at least no model not breaking artificially this symmetry. We present here a microscopic two-band model based on the structure of energetic levels of holes in CuO2 conducting layers of cuprates. In particular, our Hamiltonian does not contain ad hoc terms implying – explicitly – different masses for electrons and holes. We prove that two energetically near-lying interacting bands can explain the electron-hole asymmetry. Indeed, we rigorously analyze the phase diagram of the model and show that the critical temperatures for fermion densities below half-filling can manifest a very different behavior as compared to the case of densities above half-filling. This fact results from the inter-band interaction and intra-band Coulomb repulsion in interplay with thermal fluctuations between two energetic levels. So, if the energy difference b...

Inhomogeneous Fermi and quantum spin systems on lattices

We study the thermodynamic properties of a certain type of space-inhomogeneous Fermi and quantum spin systems on lattices. We are particularly interested in the case where the space scale of the inhomogeneities stays macroscopic, but very small as compared to the side-length of the box containing fermions or spins. The present study is however not restricted to “macroscopic inhomogeneities” and also includes the (periodic) microscopic and mesoscopic cases. We prove that – as in the homogeneous case – the pressure is, up to a minus sign, the conservative value of a two-person zero-sum game, named here thermodynamic game. Because of the absence of space symmetries in such inhomogeneous systems, it is not clear from the beginning what kind of object equilibrium states should be in the thermodynamic limit. However, we give rigorous statements on correlations functions for large boxes.

Decay of Complex-Time Determinantal and Pfaffian Correlation Functionals in Lattices

We supplement the determinantal and Pfaffian bounds of Sims and Warzel (Commun Math Phys 347:903–931, 2016) for many-body localization of quasi-free fermions, by considering the high dimensional case and complex-time correlations. Our proof uses the analyticity of correlation functions via the Hadamard three-line theorem. We show that the dynamical localization for the one-particle system yields the dynamical localization for the many-point fermionic correlation functions, with respect to the Hausdorff distance in the determinantal case. In Sims and Warzel (2016), a stronger notion of decay for many-particle configurations was used but only at dimension one and for real times. Considering determinantal and Pfaffian correlation functionals for complex times is important in the study of weakly interacting fermions.

Algebraic Quantum Mechanics

The main principles of physics were considered as well-founded by the end of the nineteenth century, even with, for instance, no satisfactory explanation of the phenomenon of thermal radiation, first discovered in 1860 by G. Kirchhoff.

Lieb–Robinson Bounds for Non-autonomous Dynamics

Like in Sect. 4, we only consider fermion systems, but all results can easily be extended to quantum spin systems (Sect. 3.6). For quantum spin systems, note that Lieb–Robinson bounds for non-autonomous dynamics have already been considered in [BMNS]. However, [BMNS] only proves Lieb–Robinson bounds for commutators, while the multi-commutator case was not considered, in contrast with results of this section. Observe also that some aspects of the non-autonomous case can be treated in a similar way to the autonomous case. However, several important arguments cannot be directly extended to the non-autonomous situation. Here, we only address in detail the technical issues which are specific to the non-autonomous problem. See for instance Corollary 5.2 (iii), Lemma 5.3, Theorems 5.5, and 5.7.

Applications to Conductivity Measures

Altogether, the classical theory of linear conductivity (including the theory of (Landau) Fermi liquids, see, e.g., [BP4] for a historical perspective) is more like a makeshift theoretical construction than a smooth and complete theory.

Algebraic Setting for Interacting Fermions on the Lattice

All quantum particles carry an intrinsic form of angular momentum, the so–called spin, first introduced by W. Pauli in the twenties.