Equilibrium and nonequilibrium entanglement properties of 2D and 3D Fermi gases

We investigate the entanglement properties of the equilibrium and nonequilibrium quantum dynamics of 2D and 3D Fermi gases, by computing entanglement entropies of extended space regions, which generally show multiplicative logarithmic corrections to the leading power-law behaviors, corresponding to the logarithmic corrections to the area law. We consider 2D and 3D Fermi gases of N particles constrained within a limited space region, for example by a hard-wall trap, at equilibrium at T=0, i.e. in their ground state, and compute the first few terms of the asymptotic large-N behaviors of entanglement entropies and particle fluctuations of subsystems with some convenient geometries, which allow us to significantly extend their computation. Then, we consider their nonequilibrium dynamics after instantaneously dropping the hard-wall trap, which allows the gas to expand freely. We compute the time dependence of the von Neumann entanglement entropy of space regions around the original trap. We show that at small time it is characterized by the relation $S \approx \pi^2 V/3$ with the particle variance, and multiplicative logarithmic corrections to the leading power law, i.e. $S \sim t^{1-d}\ln(1/t)$.Comment: 14 pages, added some ref

Andreev-Bashkin effect in superfluid cold gases mixture

We study a mixture of two superfluids with density-density and current-current (Andreev-Bashkin) interspecies interactions. The Andreev-Bashkin coupling gives rise to a dissipationless drag (or entrainment) between the two superfluids. Within the quantum hydrodynamics approximation, we study the relations between speeds of sound, susceptibilities and static structure factors, in a generic model in which the density and spin dynamics decouple. Due to translational invariance, the density channel does not feel the drag. The spin channel, instead, does not satisfy the usual Bijl-Feynman relation, since the f-sum rule is not exhausted by the spin phonons. The very same effect on one dimensional Bose mixtures and their Luttinger liquid description is analysed within perturbation theory. Using diffusion quantum Monte Carlo simulations of a system of dipolar gases in a double layer configuration, we confirm the general results. Given the recent advances in measuring the counterflow instability, we also study the effect of the entrainment on the dynamical stability of a superfluid mixture with non-zero relative velocity.Comment: 12 pages, 4 figure

Finite-size scaling at first-order quantum transitions

We study finite-size effects at first-order quantum transitions (FOQTs). We show that the low-energy properties show a finite-size scaling (FSS) behavior, the relevant scaling variable being the ratio of the energy associated with the perturbation driving the transition and the finite-size energy gap at the FOQT point. The size dependence of the scaling variable is therefore essentially determined by the size dependence of the gap at the transition, which in turn depends on the boundary conditions. Our results have broad validity and, in particular, apply to any FOQT characterized by the degeneracy and crossing of the two lowest-energy states in the infinite-volume limit. In this case, a phenomenological two-level theory provides exact expressions for the scaling functions. Numerical results for the quantum Ising chain in transverse and parallel magnetic fields support the FSS ansatzes.Comment: 5 page

Bose-Einstein condensation and critical behavior of two-component bosonic gases

We study Bose-Einstein condensation (BEC) in three-dimensional two-component bosonic gases, characterizing the universal behaviors of the critical modes arising at the BEC transitions. For this purpose, we use field-theoretical (FT) renormalization-group (RG) methods and perform mean-field and numerical calculations. The FT RG analysis is based on the Landau-Ginzburg-Wilson Phi4 theory with two complex scalar fields which has the same symmetry as the bosonic system. In particular, for identical bosons with exchange Z_2,e symmetry, coupled by effective density-density interactions, the global symmetry is Z_2e X U(1) X U(1). At the BEC transition it may break into Z_2,e X Z_2 X Z_2 when both components condense simultaneously, or to U(1) X Z_2 when only one component condenses. This implies different universality classes for the corresponding critical behaviors. Numerical simulations of the two-component Bose-Hubbard model in the hard-core limit support the RG prediction: when both components condense simultaneously, the critical behavior is controlled by a decoupled XY fixed point, with unusual slowly-decaying scaling corrections arising from the on-site inter-species interaction.Comment: 13 page

Scaling behaviour of quantum systems at thermal and quantum phase transitions

Experimental setups are finite in space and hardly ever in homogeneous conditions. This is very different from the ideal settings of the thermodynamic limit often adopted in condensed matter theories. Therefore, close to phase transitions, where typically long range correlations build up, it is important to correctly take into account the way in which boundaries and inhomogeneities affect the critical behaviour. This can be achieved by means of the finite-size (FSS) and trap-size (TSS) scaling theories, which generally apply to continuous phase transitions, where one can define a diverging length scale. FSS and TSS are reviewed in the first part of this work, together with some general properties of systems close to phase transitions. We then numerically study the TSS properties of the continuous finite-temperature phase transition of the Bose-Hubbard model (BH) in two and three dimension. This quantum model realistically describes experiments with ultra-cold bosonic gases trapped in optical lattices. In three dimensions, the BH exhibits a standard normal-to-superfluid transition. In two dimensions, the transition becomes of the Berezinski-Kosterlitz-Thouless type, characterised by logarithmic corrections to scaling. We perform thorough FSS analyses of quantum Monte Carlo data in homogeneous conditions to extract the value critical temperature. In two dimensions, this requires devising a matching method in which the FSS behaviour of the 2D BH is matched to the classical 2D XY model, whose transition belongs to the same universality class. We subsequently verify the validity of the TSS ansatz by simulating the trapped systems at the critical temperature. We find that the TSS theory is general and universal once one takes into account the effective way in which the trapping potential couples to the critical modes of the system. In the last part of this Thesis, we extend the FSS and TSS to discontinuous (or first order) quantum phase trnasitions. Discontinuous transitions do not develop a diverging length scale in the thermodynamic limit, but are rather characterised by the coexistence of domains of different phases at the transition. The typical size of single-phase domains induce a behaviour that closely resembles finite size scaling. We find that the scaling variable that parametrises the scaling behaviour at discontinuous transitions is the ratio of the perturbation energy driving the transition to the finite-size energy gap. We further find that inhomogeneous systems exibiting first order transitions can be treated heuristically in analogy with the TSS behaviour at continuous transitions. These findings are confirmed numerically on the quantum Ising and quantum Potts chains, which are simulated using density matrix renormalisation group techniques

Finite-size scaling at the first-order quantum transitions of quantum Potts chains

We investigate finite-size effects in quantum systems at first-order quantum transitions. For this purpose we consider the one-dimensional q-state Potts models which undergo a first-order quantum transition for any q>4, separating the quantum disordered and ordered phases with a discontinuity in the energy density of the ground state. The low-energy properties around the transition show finite-size scaling, described by general scaling ansatzes with respect to appropriate scaling variables. The size dependence of the scaling variables presents a particular sensitiveness to boundary conditions, which may be considered as a peculiar feature of first-order quantum transitions.Comment: 10 page

Scaling phenomena driven by inhomogeneous conditions at first-order quantum transitions

We investigate the effects of smooth inhomogeneities at first-order quantum transitions (FOQT), such as those arising from the presence of a space-dependent external field, which smooths out the typical discontinuities of the low-energy properties. We argue that scaling phenomena develop at the transition region where the external field takes the value corresponding to the FOQT of the homogenous system. We present numerical evidence of such scaling phenomena at the FOQTs of quantum Ising chains, driven by a parallel magnetic field when the system is in the ferromagnetic phase, and at the FOQT of the q-state Potts chain for q>4, driven by an even temperature-like parameter giving rise to a discontinuity of the ground-state energy density.Comment: 11 page

Direct observation of incommensurate magnetism in Hubbard chains

The interplay between magnetism and doping is at the origin of exotic strongly correlated electronic phases and can lead to novel forms of magnetic ordering. One example is the emergence of incommensurate spin-density waves with a wave vector that does not match the reciprocal lattice. In one dimension this effect is a hallmark of Luttinger liquid theory, which also describes the low energy physics of the Hubbard model. Here we use a quantum simulator based on ultracold fermions in an optical lattice to directly observe such incommensurate spin correlations in doped and spin-imbalanced Hubbard chains using fully spin and density resolved quantum gas microscopy. Doping is found to induce a linear change of the spin-density wave vector in excellent agreement with Luttinger theory predictions. For non-zero polarization we observe a decrease of the wave vector with magnetization as expected from the Heisenberg model in a magnetic field. We trace the microscopic origin of these incommensurate correlations to holes, doublons and excess spins which act as delocalized domain walls for the antiferromagnetic order. Finally, when inducing interchain coupling we observe fundamentally different spin correlations around doublons indicating the formation of a magnetic polaron