Starting from the 3D Gross-Pitaevskii equation and using a variational approach, we derive an effective 1D wave-equation that describes the axial dynamics of a Bose condensate confined in an external potential with cylindrical symmetry. The trapping potential is harmonic in the transverse direction and generic in the axial one. Our equation, that is a time-dependent non-polynomial nonlinear Schr\"odinger equation (1D NPSE), can be used to model cigar-shaped condensates, whose dynamics is essentially 1D. We show that 1D NPSE gives much more accurate results than all other effective equations recently proposed. By using 1D NPSE we find analytical solutions for bright and dark solitons, which generalize the ones known in the literature. We deduce also an effective 2D non-polynomial Schr\"odinger equation (2D NPSE) that models disc-shaped Bose condensates confined in an external trap that is harmonic along the axial direction and generic in the transverse direction. In the limiting cases of weak and strong interaction, our approach gives rise to Schr\"odinger-like equations with different polynomial nonlinearities

Thermophoresis is particle motion induced by thermal gradients. Akin to other driven transport processes, such as the Soret effect in simple fluid mixtures, or electrophoresis and diffusiophoresis in colloidal suspensions, it is, both experimentally and theoretically, a challenging subject. Rather than being a comprehensive recollection, this review aims to be a critical re-examination of the experimental and theoretical tools used to investigate thermophoresis, and of some recent relevant results that may unravel novel aspects of colloid solvation forces. The perspectives of thermophoresis as a tool for particle manipulation in microfluidics are also emphasized. Contents 16 6. Conclusions and perspectives 17 References 17

By direct calculations of the spin gap in the frustrated Heisenberg model on the square lattice, with nearest-(J1) and next-nearest-neighbor (J2) super-exchange couplings, we provide a solid evidence that the spin-liquid phase in the frustrated regime 0.45 J2/J1 0.6 is gapless. Our numerical method is based on a variational wave function that is systematically improved by the application of few Lanczos steps and allows us to obtain reliable extrapolations in the thermodynamic limit. The peculiar nature of the non-magnetic state is unveiled by the existence of S = 1 gapless excitations at k = (π, 0) and (0, π). The magnetic transition can be described and interpreted by a variational state that is built from Abrikosov fermions having a Z2 gauge structure and four Dirac points in the spinon spectrum. PACS numbers: 1/L J 2 /J 1 =0.3 J 2 /J 1 =0.4 J 2 /J 1 =0.45 J 2 /J 1 =0.5 J 2 /J 1 =0.55 J 2 /J 1 =0.45, p=0 J 2 /J 1 =0.45, variance extrapolation J 2 /J 1 =0.5, p=0 J 2 /J 1 =0.5, variance extrapolation

No abstract

The Resonating Valence Bond (RVB) theory for two-dimensional quantum antiferromagnets is shown to be the correct paradigm for large enough "quantum frustration". This scenario, proposed long time ago but never confirmed by microscopic calculations, is very strongly supported by a new type of variational wave function, which is extremely close to the exact ground state of the J1−J2 Heisenberg model for 0.4 < ∼ J2/J1 < ∼ 0.5. This wave function is proposed to represent the generic spin-half RVB ground state in spin liquids. 75.10.Jm, 71.27.+a, 74.20.Mn The question whether a frustrated spin-half system is well described by a spin-liquid ground state (GS) -with no type of crystalline order -25 years after the first proposal [1] is still controversial, mainly because of the lack of reliable analytical or numerical solutions of model systems. For unfrustrated or weakly frustrated quantum antiferromagnets a deep understanding of the nature of the GS together with a quantitative description of the ordered phase is obtained by including Gaussian quantum fluctuations over a classical Néel state. [2,3] For sizeable frustration, instead, this description is known to break down. However, the short-range RVB state [4] does not prove a good starting point for the description of frustrated models; it rather turns out to be the exact GS of ad hoc Hamiltonians. [4][5][6] As a prototype of a realistic frustrated two-dimensional system, which has been recently realized experimentally in Li 2 VOSiO 4 compounds, [7] we investigate the spinhalf Heisenberg model with nearest (J 1 ) and next-nearest neighbor (J 2 ) superexchange couplings:on an N −site square lattice with periodic boundary conditions. In the (J 2 = 0) unfrustrated case, it is well established that the GS of the Heisenberg Hamiltonian has Néel long-range order, with a sizable value of the antiferromagnetic order parameter.[8] However, variational studies [9] have shown that disordered, long-range RVB states have energies very close to the exact one. It is therefore natural to imagine that by turning on the next-nearest neighbor interaction J 2 , the combined effect of frustration and zero-point motion may eventually melt antiferromagnetism and stabilize a non-magnetic GS of purely quantum-mechanical nature. Indeed, for 0.4 < ∼ J 2 /J 1 < ∼ 0.6 there is a general consensus on the disappearance of the Néel order towards a state whose nature is still much debated. [10] In a seminal paper, [11] Anderson proposed that a physically transparent description of a RVB state can be obtained in fermionic representation by starting from a BCS-type pairing wave function (WF), of the form

Using computational techniques, it is shown that pairing is a robust property of hole doped antiferromagnetic (AF) insulators. In one dimension (1D) and for two-leg ladder systems, a BCS-like variational wave function with long-bond spin-singlets and a Jastrow factor provides an accurate representation of the ground state of the t-J model, even though strong quantum fluctuations destroy the off-diagonal superconducting (SC) long-range order in this case. However, in two dimensions (2D) it is argued -and numerically confirmed using several techniques, especially quantum Monte Carlo (QMC) -that quantum fluctuations are not strong enough to suppress superconductivity. 74.20.Mn, 71.10.Fd, 71.10.Pm, 71.27.+a The nature of high temperature superconductors remains an important unsolved problem in condensed matter physics. Strong electronic correlations are widely believed to be crucial for the understanding of these materials. Among the several proposed theories are those where antiferromagnetism induces pairing in the d x 2 −y 2 channel [1]. These approaches include the following two classes: (i) theories based on Resonant Valence Bond (RVB) wave functions, with electrons paired in long spin singlets in all possible arrangements [2,3], and (ii) theories based on two-hole d x 2 −y 2 bound states at infinitesimal doping, formed to minimize the damage of individual holes to the AF order parameter, which condense at finite pair density into a superconductor [4]. However, recent density matrix renormalization group (DMRG) calculations have seriously questioned these approaches since non-SC striped ground states were reported for realistic couplings and densities in the t-J model [5]. Clearly to make progress in the understanding of copper oxides, the 2D t-J model ground state must be fully understood, to distinguish among the many proposals.In this paper, using a variety of powerful numerical techniques, the properties of the t-J model are investigated. Our main result is that in the realistic regime of couplings the 2D t-J model supports a d x 2 −y 2 SC ground state, confirming theories of Cu-oxides based on AF correlations. The t-J model used here is. . stands for nearestneighbor sites, and n i and S i are the electron density and spin at site i, respectively. Our study focuses on the low hole-doping region of chains, two-leg ladders, and square clusters, using different numerical techniques: QMC (pure variational and fixed-node (FN) approximations), DMRG, and Lanczos. Within our QMC approach, it is possible to further improve the variational and FN accuracy by applying a few (p ≤ 2) Lanczos steps to the variational (p = 0) wave function |Ψ V ,. Non-variational estimates of energy and correlation functions can also be extracted with the variance-extrapolation method [6].Our BCS variational wave function is defined as

We investigate the dynamics of Bose-condensed bright solitons made of alkali-metal atoms with negative scattering length and under harmonic confinement in the transverse direction. Contrary to the 1D case, the 3D bright soliton exists only below a critical attractive interaction which depends on the extent of confinement. Such a behavior is also found in multi-soliton condensates with box boundary conditions. We obtain numerical and analytical estimates of the critical strength beyond which the solitons do not exist. By using an effective 1D nonpolynomial nonlinear Schr\"odinger equation (NPSE), which accurately takes into account the transverse dynamics of cigar-like condensates, we numerically simulate the dynamics of the "soliton train" reported in a recent experiment (Nature {\bf 417} 150 (2002)). Then, analyzing the macroscopic quantum tunneling of the bright soliton on a Gaussian barrier we find that its interference in the tunneling region is strongly suppressed with respect to non-solitonic case; moreover, the tunneling through a barrier breaks the solitonic nature of the matter wave. Finally, we show that the collapse of the soliton is induced by the scattering on the barrier or by the collision with another matter wave when the density reaches a critical value, for which we derive an accurate analytical formula

We study the formation of bright solitons in a Bose-Einstein condensate of 7Li atoms induced by a sudden change in the sign of the scattering length from positive to negative, as reported in a recent experiment [Nature (London) 417, 150 (2002)]]. The numerical simulations are performed by using the Gross-Pitaevskii equation with a dissipative three-body term. We show that a number of bright solitons is produced and this can be interpreted in terms of the modulational instability of the time-dependent macroscopic wave function of the Bose condensate. In particular, we derive a simple formula for the number of solitons that is in good agreement with the numerical results. We find that during the motion of the soliton train in an axial harmonic potential the number of solitonic peaks changes in time and the density of individual peaks shows an intermittent behavior.