Asymptotic lower bound for the gap of Hermitian matrices having ergodic ground states and infinitesimal off-diagonal elements

Given a $M\times M$ Hermitian matrix $\mathcal{H}$ with possibly degenerate eigenvalues $\mathcal{E}_1 < \mathcal{E}_2 < \mathcal{E}_3< \dots$, we provide, in the limit $M\to\infty$, a lower bound for the gap $\mu_2 = \mathcal{E}_2 - \mathcal{E}_1$ assuming that (i) the eigenvector (eigenvectors) associated to $\mathcal{E}_1$ is ergodic (are all ergodic) and (ii) the off-diagonal terms of $\mathcal{H}$ vanish for $M\to\infty$ more slowly than $M^{-2}$. Under these hypotheses, we find $\varliminf_{M\to\infty} \mu_2 \geq \varlimsup_{M\to\infty} \min_{n} \mathcal{H}_{n,n}$. This general result turns out to be important for upper bounding the relaxation time of linear master equations characterized by a matrix equal, or isospectral, to $\mathcal{H}$. As an application, we consider symmetric random walks with infinitesimal jump rates and show that the relaxation time is upper bounded by the configurations (or nodes) with minimal degree.Comment: 5 page

Perturbative unitarity bounds for effective composite models

In this paper we present the partial wave unitarity bound in the parameter space of dimension-5 and dimension-6 effective operators that arise in a compositeness scenario. These are routinely used in experimental searches at the LHC to constraint contact and gauge interactions between ordinary Standard Model fermions and excited (composite) states of mass $M$. After deducing the unitarity bound for the production process of a composite neutrino, we implement such bound and compare it with the recent experimental exclusion curves for Run 2, the High-Luminosity and High-Energy configurations of the LHC. Our results also applies to the searches where a generic single excited state is produced via contact interactions. We find that the unitarity bound, so far overlooked, is quite complelling and significant portions of the parameter space ($M,\Lambda$) become excluded in addition to the standard request $M \le \Lambda$.Comment: This version of the paper merges the previous version published in Phys. Lett. B 795 (2019) 644-649 (https://doi.org/10.1016/j.physletb.2019.06.042) with the subsequent Erratum currently in press in Physics Letters B (https://doi.org/10.1016/j.physletb.2019.134990

Exact ground state for a class of matrix Hamiltonian models: quantum phase transition and universality in the thermodynamic limit

By using a recently proposed probabilistic approach, we determine the exact ground state of a class of matrix Hamiltonian models characterized by the fact that in the thermodynamic limit the multiplicities of the potential values assumed by the system during its evolution are distributed according to a multinomial probability density. The class includes i) the uniformly fully connected models, namely a collection of states all connected with equal hopping coefficients and in the presence of a potential operator with arbitrary levels and degeneracies, and ii) the random potential systems, in which the hopping operator is generic and arbitrary potential levels are assigned randomly to the states with arbitrary probabilities. For this class of models we find a universal thermodynamic limit characterized only by the levels of the potential, rescaled by the ground-state energy of the system for zero potential, and by the corresponding degeneracies (probabilities). If the degeneracy (probability) of the lowest potential level tends to zero, the ground state of the system undergoes a quantum phase transition between a normal phase and a frozen phase with zero hopping energy. In the frozen phase the ground state condensates into the subspace spanned by the states of the system associated with the lowest potential level.Comment: 31 pages, 13 figure

Analytical probabilistic approach to the ground state of lattice quantum systems: exact results in terms of a cumulant expansion

We present a large deviation analysis of a recently proposed probabilistic approach to the study of the ground-state properties of lattice quantum systems. The ground-state energy, as well as the correlation functions in the ground state, are exactly determined as a series expansion in the cumulants of the multiplicities of the potential and hopping energies assumed by the system during its long-time evolution. Once these cumulants are known, even at a finite order, our approach provides the ground state analytically as a function of the Hamiltonian parameters. A scenario of possible applications of this analyticity property is discussed.Comment: 26 pages, 5 figure

Signatures of macroscopic quantum coherence in ultracold dilute Fermi gases

We propose a double-well configuration for optical trapping of ultracold two-species Fermi-Bose atomic mixtures. Two signatures of macroscopic quantum coherence attributable to a superfluid phase transition for the Fermi gas are analyzed. The first signature is based upon tunneling of Fermi pairs when the power of the deconfining laser beam is significantly reduced. The second relies on the observation of interference fringes in a regime where the fermions are trapped in two sharply separated minima of the potential. Both signatures rely on small decoherence times for the Fermi samples, which should be possible by reaching low temperatures using a Bose gas as a refrigerator, and a bichromatic optical dipole trap for confinement, with optimal heat-capacity matching between the two species

Cooling dynamics of ultracold two-species Fermi-Bose mixtures

We compare strategies for evaporative and sympathetic cooling of two-species Fermi-Bose mixtures in single-color and two-color optical dipole traps. We show that in the latter case a large heat capacity of the bosonic species can be maintained during the entire cooling process. This could allow to efficiently achieve a deep Fermi degeneracy regime having at the same time a significant thermal fraction for the Bose gas, crucial for a precise thermometry of the mixture. Two possible signatures of a superfluid phase transition for the Fermi species are discussed.Comment: 4 pages, 3 figure

An exact representation of the fermion dynamics in terms of Poisson processes and its connection with Monte Carlo algorithms

We present a simple derivation of a Feynman-Kac type formula to study fermionic systems. In this approach the real time or the imaginary time dynamics is expressed in terms of the evolution of a collection of Poisson processes. A computer implementation of this formula leads to a family of algorithms parametrized by the values of the jump rates of the Poisson processes. From these an optimal algorithm can be chosen which coincides with the Green Function Monte Carlo method in the limit when the latter becomes exact.Comment: 4 pages, 1 PostScript figure, REVTe

Erratum to: “Perturbative unitarity bounds for effective composite models” [Phys. Lett. B 795 (2019) 644-649]

Numerical results for the partial wave unitarity bounds on the parameter space (Lambda, M) of dimension-6 effective operators of a composite scenario presented in Biondini et al. (2019) [1] are revised. Figs. 2-5 and Table 1 are to be replaced by the following corresponding figures and table. We briefly comment on the impact on the conclusions presented in the original article

Non-Commutativity effects in the Dirac equation in crossed electric and magnetic fields

In this paper we present exact solutions of the Dirac equation on the non-commutative plane in the presence of crossed electric and magnetic fields. In the standard commutative plane such a system is known to exhibit contraction of Landau levels when the electric field approaches a critical value. In the present case we find exact solutions in terms of the non-commutative parameters $\eta$ (momentum non-commutativity) and $\theta$ (coordinate non-commutativity) and provide an explicit expression for the Landau levels. We show that non-commutativity preserves the collapse of the spectrum. We provide a dual description of the system: (i) one in which at a given electric field the magnetic field is varied and the other (ii) in which at a given magnetic field the electric field is varied. In the former case we find that momentum non-commutativity ($\eta$) splits the critical magnetic field into two critical fields while coordinates non-commutativity ($\theta$) gives rise to two additional critical points not at all present in the commutative scenario.Comment: 6 pages, 4 figures, Accepted for publication by EuroPhysics Letters (EPL