33 research outputs found
Scaling phenomena driven by inhomogeneous conditions at first-order quantum transitions
No full textWe investigate the effects of smooth inhomogeneities at first-order quantum transitions (FOQTs), such as those arising in the presence of a space-dependent external field, which smooths out the discontinuities of the low-energy properties at the transition. We argue that a universal scaling behavior emerges in the space transition region close to the point in which the external field takes the value for which the homogeneous system undergoes the FOQT. We verify the general theory in two model systems. We consider the quantum Ising chain in the ferromagnetic phase and the q-state Potts chain for q = 10, investigating the scaling behavior which arises in the presence of an additional inhomogeneous parallel and transverse magnetic field, respectively. Numerical results are in full agreement with the general theory
Finite-Size Scaling at First-Order Quantum Transitions
No full textWe 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
Universal behavior of two-dimensional bosonic gases at Berezinskii-Kosterlitz-Thouless transitions
No full textWe study the universal critical behavior of two-dimensional (2D) lattice
bosonic gases at the Berezinskii-Kosterlitz-Thouless (BKT) transition, which
separates the low-temperature superfluid phase from the high-temperature normal
phase. For this purpose, we perform quantum Monte Carlo simulations of the
hard-core Bose-Hubbard (BH) model at zero chemical potential. We determine the
critical temperature by using a matching method that relates finite-size data
for the BH model with corresponding data computed in the classical XY model. In
this approach, the neglected scaling corrections decay as inverse powers of the
lattice size L, and not as powers of 1/lnL, as in more standard approaches,
making the estimate of the critical temperature much more reliable. Then, we
consider the BH model in the presence of a trapping harmonic potential, and
verify the universality of the trap-size dependence at the BKT critical point.
This issue is relevant for experiments with quasi-2D trapped cold atoms
Equilibrium and nonequilibrium entanglement properties of 2D and 3D Fermi gases
No full textWe 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 with the particle variance,
and multiplicative logarithmic corrections to the leading power law, i.e.
Phase diagram and multicritical behaviors of mixtures of three-dimensional bosonic gases
No full textWe investigate the Bose-Einstein condensation (BEC) patterns, the critical and multicritical behaviors of three-dimensional mixtures of bosonic gases with density-density interactions, characterized by a global U(1) U(1) symmetry [one U(1) transformation for each species]. In particular, we consider the three-dimensional Bose-Hubbard model for two lattice bosonic gases coupled by an on-site interspecies density-density interaction. We study the phase diagram and the critical behaviors along the transition lines of the BEC of one or both species. We present mean-field calculations and finite-size scaling analyses of quantum Monte Carlo data. We also investigate the nature of the multicritical points where the BEC transition lines of the two species meet. The corresponding universality classes are inferred from a renormalization-group analysis of the corresponding multicritical U(1)U(1) Landau-Ginzburg-Wilson Φ4 theory. We find two distinct critical behaviors, associated with bicritical and tetracritical points, respectively, depending on the relative strength of the interspecies and intraspecies interactions
Finite-size scaling at the first-order quantum transitions of quantum Potts chains
No full textWe investigate finite-size effects at first-order quantum transitions. For this purpose we consider the onedimensional
q-state quantum Potts chain, in particular with q = 10, which undergoes a first-order transition,
separating the quantum disordered and ordered phases with a discontinuity in the energy density of the ground
state. In agreement with the general theory, around the transition the low-energy properties show finite-size
scaling with respect to appropriate scaling variables. Their size dependence is particularly sensitive to boundary
conditions, which is a specific feature of first-order quantum transitions. Finally, we also discuss the finite-size
behavior of the q-state Potts model (q 2) at the first-order transitions driven by a parallel magnetic field,
occurring in the ferromagnetic phase
Bose-Einstein condensation and critical behavior of two-component bosonic gases
No full textWe study Bose-Einstein condensation (BEC) in three-dimensional two-component bosonic gases, character-
izing 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 ! 4 theory with two complex scalar fields which
has the same symmetry as the bosonic system. In particular, for identical bosons with exchange Z2 symmetry,
coupled by effective density-density interactions, the global symmetry is Z2,e ⊗ U(1) ⊗ U(1). At the BEC
transition, it may break into Z2 ,e ⊗ Z2 ⊗ Z2 when both components condense simultaneously, or to U(1)
⊗ Z2 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 onsite interspecies
interaction
Scaling behaviour of quantum systems at thermal and quantum phase transitions
No full textExperimental 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
Entanglement entropies in many dimensional Fermi gases
No full textEntanglement is a key aspect of quantum mechanics, and arguably the clearest manifestation of the non-locality that is so intimately written within the quantum theory.
The wave functions used to describe matter are, in fact, spatially extended; only when the position of a particle is measured the wave function collapses and the local particle description of matter can be used.
The counter intuitive nature of entanglement has intrigued physicists since the very beginning of the quantum era, and sparked controversies and paradoxes. Among these, we cite the famous paradox proposed by A.Einstein, B.Podolsky and N.Rosen 1935, in which a pair of entangled particles deceptively give the impression that information can be transmitted faster than the speed of light.
While in the past the concept of quantum entanglement was primarily applied to systems with few degrees of freedom, this has changed in the modern days. The developments of the theories of quantum decoherence, condensed matter and quantum information, together with the huge improvements in the experimental techniques of ultra-cold atoms, nano technologies and quantum optics, have fuelled a new interest on the entanglement of many-body systems.
An extensive literature has built up on the problem of measuring the entanglement of a part of an extended system with the rest. The interest in it was motivated, among other reasons, by the ideas of Bekenstein-Hawking black hole entropy and the holographic principle first proposed by 't Hooft. The development of measures such as the von Neumann and Rényi bipartite entanglement entropies are great theoretical achievements in these respects.
In thermodynamics, entropy is an extensive quantity, and hence scales with the volume of the system, i.e. , with the typical dimension of a system and its spatial dimension.
Entanglement entropies are somewhat peculiar in that they typically scale as the extension of the hyper-surface separating the two parts of the system one considers, i.e. . This is commonly referred in literature as the area law scaling of entanglement entropies. An extremely important application of area laws is their use as flags for quantum criticality: in fact, under reasonable assumptions, the scaling is logarithmically violated at quantum critical points, i.e. .
Measuring entanglement is often not an easy task, as a quite deep understanding of the system is required.
Spin chain models are particularly useful to overcome these limitations, in that they are sufficiently complex to have real-world counterparts and at the same time sufficiently simple to allow a rigorous, albeit sometimes only approximate, analytical solution.
By exploiting the known solution to some models in low spatial dimensions, it was possible to analytically compute entanglement entropies, both for finite-size systems and in the thermodynamic limit.
This is even more the case for some one-dimensional models at quantum critical points, whenever the system is invariant under translations, rotations and scaling transformations. In these cases the system is usually invariant under the wider conformal group in 1+1 dimensions (the second dimension being the imaginary time obtained by a Wick rotation of the real time). The powerful methods of Conformal Field Theory confirmed and sometimes extended the results on the bipartite entanglement entropies of critical one-dimensional systems.
By using these methods, together with profound mathematical theorems, it was possible to also obtain the behaviour of the entanglement and the scaling of the leading corrections in dimensions higher than one.
In this thesis we review in detail the main aspects of quantum entanglement described above.
We then build upon exact results for systems of free spinless Fermi gases in a finite-size one-dimensional box.
From these, we derive analytical expressions for the entanglement entropies of free Fermi gases in higher dimensions for the particular strip-like geometry. The results are asymptotic for the limit of a large number of particles.
Contrarily to previously published results, in this work the subleading corrections are rigorously calculated and the next order corrections can be estimated.
The formulae are thoroughly verified against numerical results in two and three dimensions, and are in perfect agreement with the theoretical predictions.
Furthermore, an argument is found for the wide oscillation of entanglement entropies that are present in dimensions higher than one. These are found to be linked to the particular way in which the Fermi sea is filled as the particle number is increased. In two dimensions, it is numerically proven that the oscillation exactly correspond to the pseudo-periodicity of perfect squares.
Finally, the most recent developments in the field of entanglement entropies for Fermi gases are reviewed, with particular attention to dynamic situations with non-trivial interactions. These, together with known relations between entanglement entropies and particle densities, provide us with a complete picture that can be tested experimentally. Some known results of experiments with ultra-cold atoms in magneto-optic traps are reported and discussed
Andreev-Bashkin effect in superfluid cold gases mixtures
No full text© 2017 The Author(s). Published by IOP Publishing Ltd on behalf of Deutsche Physikalische Gesellschaft. 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.Postprint (author's final draft
