1,380 research outputs found
Wigner Crystallization of Electrons in a One-Dimensional Lattice: A Condensation in the Space of States
No full textWe study the ground state of a system of spinless electrons interacting through a screened Coulomb potential in a lattice ring. By using analytical arguments, we show that, when the effective interaction compares with the kinetic energy, the system forms a Wigner crystal undergoing a first-order quantum phase transition. This transition is a condensation in the space of the states and belongs to the class of quantum phase transitions discussed in [M. Ostilli and C. Presilla, J. Phys. A 54, 055005 (2021).JPAMB51751-811310.1088/1751-8121/aba144]. The transition takes place at a critical value rsc of the usual dimensionless parameter rs (radius of the volume available to each electron divided by effective Bohr radius) for which we are able to provide rigorous lower and upper bounds. For large screening length these bounds can be expressed in a closed analytical form. Demanding Monte Carlo simulations allow to estimate rsc≃2.3±0.2 at lattice filling 3/10 and screening length 10 lattice constants. This value is well within the rigorous bounds 0.7≤rsc≤4.3. Finally, we show that if screening is removed after the thermodynamic limit has been taken, rsc tends to zero. In contrast, in a bare unscreened Coulomb potential, Wigner crystallization always takes place as a smooth crossover, not as a quantum phase transition
First-order quantum phase transitions as condensations in the space of states
No full textWe demonstrate that a large class of first-order quantum phase transitions, namely, transitions in which the ground state energy per particle is continuous but its first order derivative has a jump discontinuity, can be described as a condensation in the space of states. Given a system having Hamiltonian H = K + gV, where K and V are two non commuting operators acting on the space of states F, we may always write F = Fcond Fnorm where Fcond is the subspace spanned by the eigenstates of V with minimal eigenvalue and Fnorm = Fcond. If, in the thermodynamic limit, Mcond/M → 0, where M and Mcond are, respectively, the dimensions of F and Fcond, the above decomposition of F becomes effective, in the sense that the ground state energy per particle of the system, ε, coincides with the smaller between εcond and εnorm, the ground state energies per particle of the system restricted to the subspaces Fcond and Fnorm, respectively: ε = min{εcond, εnorm}. It may then happen that, as a function of the parameter g, the energies εcond and εnorm cross at g = gc. In this case, a first-order quantum phase transition takes place between a condensed phase (system restricted to the small subspace Fcond) and a normal phase (system spread over the large subspace Fnorm). Since, in the thermodynamic limit, Mcond/M → 0, the confinement into Fcond is actually a condensation in which the system falls into a ground state orthogonal to that of the normal phase, something reminiscent of Anderson's orthogonality catastrophe (Anderson 1967 Phys. Rev. Lett. 18 1049). The outlined mechanism is tested on a variety of benchmark lattice models, including spin systems, free fermions with non uniform fields, interacting fermions and interacting hard-core bosons
Residui
No full textSingolarità e singolarità isolate. Residuo di una funzione in una sin- golarità isolata. Teorema dei residui. Teorema dei residui con il residuo all’infinito. Classificazione delle singolarità isolate: singolarità eliminabili, poli di ordine m, sin- golarità essenziali. Condizione necessaria e sufficiente affinché un punto singolare isolato di una funzione analitica sia un polo di ordine m e formula per il corrispon- dente residuo. Zeri di ordine m delle funzioni analitiche. Condizione necessaria e sufficiente affinché una funzione analitica abbia uno zero di ordine m. Teorema di identità. Gli zeri delle funzioni analitiche non costanti sono isolati e di ordine finito. Condizione sufficiente affinché una funzione del tipo f(z) = p(z)/q(z) abbia un polo di ordine m e formula per il corrispondente residuo. Comportamento di una funzio- ne analitica in prossimità delle singolarità isolate. Lemma di Riemann. Teorema di Casorati-Weierstrass
Asymptotic lower bound for the gap of Hermitian matrices having ergodic ground states and infinitesimal off-diagonal elements
No full textGivena M M Hermitian matrix H with possibly degenerate eigenvalues E11<E2<E3<...
, we provide, in the limit M →∞, a lower bound for the gap μ 2=E2−E1assum-ing that i) the eigenvector (eigenvectors) associated to
E1is ergodic (are all ergodic) and ii) theoff-diagonal terms of H vanish for M→∞. Under these hypotheses, we find lim M→∞2
≥ limM→∞minnHn,n. This general result turns out to be important for upper bounding the relax-
ation time of linear master equations characterized by a matrix equal, or isospectral, to 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 degr
Thermalization of the Lipkin-Meshkov-Glick model in blackbody radiation
No full textn a recent work, we have derived simple Lindblad-based equations for the thermalization of systems in contactwith a thermal reservoir. Here, we apply these equations to the Lipkin-Meshkov-Glick model in contact withblackbody radiation and analyze the dipole matrix elements involved in the thermalization process. We find thatthe thermalization can be complete only if the density is sufficiently high, while, in the limit of low density, thesystem thermalizes partially, namely, within the Hilbert subspaces where the total spin has a fixed value. In thisregime, and in the isotropic case, we evaluate the characteristic thermalization time analytically, and show thatit diverges with the system size in correspondence with the critical points and inside the ferromagnetic region.Quite interestingly, at zero temperature the thermalization time diverges only quadratically with the system size,whereas quantum adiabatic algorithms, aimed at finding the ground state of the same system, imply a cubicdivergence of the required adiabatic time
Phase transitions and gaps in quantum random energy models
No full textBy using a previously established exact characterization of the ground state of random potential systems in the thermodynamic limit, we determine the ground and first excited energy levels of quantum random energy models, discrete and continuous. We rigorously establish the existence of a universal first order quantum phase transition, obeyed by both the ground and the first excited states. The presence of an exponentially vanishing minimal gap at the transition is general but, quite interestingly, the gap averaged over the realizations of the random potential is finite. This fact leaves still open the chance for some effective quantum annealing algorithm, not necessarily based on a quantum adiabatic scheme
Ground state of many-body lattice systems via a central limit theorem
No full textWe review a novel approach to evaluate the ground-state properties of many-body lattice systems based on an exact probabilistic representation of the dynamics and its long time approximation via a central limit theorem. The choice of the asymptotic density probability used in the calculation is discussed in detail
Finite-temperature quantum condensations in the space of states: A different perspective on quantum annealing
No full textIn nature, everything occurs at finite temperature, and quantum phase transitions (QPTs) cannot be an exception. Nevertheless, they are still mainly discussed and formulated at zero temperature. We show that the condensation QPTs recently introduced at zero temperature can naturally be extended to finite temperature just by replacing ground-state energies with corresponding free energies. We illustrate this criterion in the paradigmatic Grover model and in a system of free fermions in a one-dimensional inhomogeneous lattice. In agreement with expected universal features, the two systems show structurally similar phase diagrams. Last, we explain how finite-temperature condensation QPTs can be used to construct quantum annealers having, at finite temperature, output probability exponentially close to 1 in the system size. As examples we consider again the Grover model and the fermionic system, the latter being well within the reach of present heterostructure technology
AUGER-SPECTRA OF 3D TRANSITION-METALS WITH ORBITAL DEGENERACY
No full textThe equation of motion approach is employed to derive the two-body Green function. relevant for the calculation of Auger spectra, in the case of systems well described by the Hubbard Hamiltonian with many bands. The results have been used to calculate the Mlvv (M, valence-valence) Auger rate of copper, which has been shown to be well described by the present theory. as distinct contributions belonging to different bands are clearly present
Ground state of many-body lattice systems: an analytical probabilistic approach
No full textOn the grounds of a Feynman–Kac-type formula for Hamiltonian lattice systems, we derive analytical expressions for the matrix elements of the evolution operator. These expressions are valid at long times when a central limit theorem applies. As a remarkable result, we find that the ground-state energy as well as all the correlation functions in the ground state are determined semi-analytically by solving a simple scalar equation. Furthermore, explicit solutions of this equation are obtained in the noninteracting case
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