The Effect of a Long-Range Correlated-Hopping Interaction on

Tao Yang, Fa-Kai Wen, Kun Hao, Like Cao, Rui‐Hong Yue, Ignazio Francesco Licata, Sauro Succi A

Published Jan 1, 2015• 0 citations

Abstract

School of Physical Science and Technology, Yangzhou University, Yangzhou 225002, China* Authors to whom correspondence should be addressed; E-Mails: yangt@nwu.edu.cn (T.Y.);rhyue@yzu.edu.cn (R.Y.); Tel.: +86-29-88303513; Fax: +86-29-88302331.Academic Editors: Ignazio Licata and Sauro SucciReceived: 16 May 2015 / Accepted: 25 August 2015 / Published: 28 August 2015Abstract: We introduce a long-range particle and spin interaction into the standard Barievmodel and show that this interaction is equivalent to a phase shift in the kinetic term ofthe Hamiltonian. When the particles circle around the chain and across the boundary,the accumulated phase shift acts as a twist boundary condition with respect to the normalperiodic boundary condition. This boundary phase term depends on the total number ofparticles in the system and also the number of particles in different spin states, whichrelates to the spin ﬂuctuations in the system. The model is solved exactly via a unitarytransformation by the coordinate Bethe ansatz. We calculate the Bethe equations and workout the energy spectrum with varying number of particles and spins.Keywords: Bariev model; long-range interaction; twist boundary condition; energyspectrum1. IntroductionOne dimensional (1D) (quasi-1D) systems exhibit some of the most diverse and intriguing physicalphenomena seen in all of condensed matter physics, such as charge (spin) density waves, quantumwires, quantum Hall bars, Josephson junction arrays, polymers and 1D Bose-Einstein condensates. Thecomplete description of a solid is a complex many body problem. The particles are strongly correlatedand cannot be understood by removing the interactions between them or by considering the effects of

Related papers

The Effect of a Long-Range Correlated-Hopping Interaction on Bariev Spin Chains

Tao Yang, Fa-Kai Wen, Kun Hao +2 more

We introduce a long-range particle and spin interaction into the standard Bariev model and show that this interaction is equivalent to a phase shift in the kinetic term of the Hamiltonian. When the particles circle around the chain and across the boundary, the accumulated phase shift acts as a twist boundary condition with respect to the normal periodic boundary condition. This boundary phase term depends on the total number of particles in the system and also the number of particles in different spin states, which relates to the spin fluctuations in the system. The model is solved exactly via a unitary transformation by the coordinate Bethe ansatz. We calculate the Bethe equations and work out the energy spectrum with varying number of particles and spins.

EntropyAug 28, 20152 citations

Pairing and topological phases in cold atoms with long range interactions

Lorenzo Gotta

Pairing and topological phases in cold atoms with long range interactions Chapter 1 Introduction The systematic study and classiﬁcation of phase transitions has become a popular research area in physics over the past decades. The most familiar phenomena associated to the concept of phase transition are the ones giving rise to macroscopic changes in the properties of strongly correlated many-body systems due to thermal ﬂuctuations. The most prominent examples of this kind of behaviour are the ferromagnetic-paramagnetic transition in lattice spin models and the liquid-gas transition. On the other hand, when dealing with quantum-mechanical systems changes of matter from one state to another can arise as the result of the variation of parameters other than temperature. In particular, when a given many-body quantum problem is studied at zero temperature, thereby suppressing thermal ﬂuctuations, it is the interplay between kinetic energy and interactions that can cause the system to jump into diﬀerent phases as the model parameters are varied. Phase transitions occurring at zero temperature are widely known as quantum phase transitions. In the framework of the physics of phase transitions, the role of dimensionality is a noticeable one. The major challenge in this perspective is the treatment of low dimensional systems, where standard mean-ﬁeld treatments and ordinary perturbation theory are known to fail due to the enhancement of ﬂuctuations. The goal of describing collective behaviours in reduced dimensionality is usually pursued in one-dimensional (1D) systems, which turn out to be simple enough to be studied in an eﬀective way through both numerical and analytical techniques, while exhibiting a rich and deeply fascinating phenomenology. As a matter of fact, since particles conﬁned to one dimension cannot avoid interactions with each other, the typical low energy excitations of the system are represented by collective density waves, which substitute nearly free quasiparticle excitations characterizing the behaviour of three-dimensional Fermi liquids in the description of the eﬀects of the interplay between kinetic ﬂuctuations and interactions. On the other hand, the tools employed in the activity of tackling 1D manybody problems are eﬃcient and well developed. The analytical techniques aremainly based on the bosonization technique, which allows for the reformulation of strongly correlated 1D systems onto free bosonic theories, which can then be analyzed by means of standard path integral techniques. Meanwhile, the density matrix renormalization group (DMRG) algorithm represents the state-of-theart numerical machinery in the study of 1D physics, since it allows to eﬃciently extract the low energy properties of the model Hamiltonian and consequently characterize the corresponding phase diagram by means of the behaviour of properly chosen order parameters. As a ﬁnal remark pointing out the relevance of the research eﬀort in the direction of a better understanding of 1D many-body quantum systems, it is worth noticing that 1D systems do not represent only a playground for theorists, but have been experimentally realized by means of setups with cold atoms trapped in optical lattices, where the experimentalist is able to tune the model parameters by varying the characteristic features of the trapping laser beams and meanwhile has a comfortable access to the most relevant observables.

Jul 26, 20190 citations

Interaction Effects

Hui Zhai

1. Define Bose-Einstein condensation in an interacting boson system in term of macroscopic occupation. 2. Introduce the Gross-Pitaevskii equation and its applicable conditions. 3. Derive the hydrodynamic equations and various physical consequences of the hydrodynamics with or without harmonic trap. 4. Introduce sound wave and emphasize the difference between the wave behavior and the diffusive behavior. 5. Introduce the criterion for superfluidity and determine the superfluid critical velocity. 6. Introduce two-fluid hydrodynamics. 7. Introduce the Bogoliubov theory. 8. Introduce various concepts related to quasi-particle, including ground state as vacuum of quasi-particles, the zero-point energy of quasi-particle, quasi-particle lifetime and vacuum fluctuation. 9. Introduce healing length as the characteristic length scale for low-energy physics of an interacting Bose condensate. 10. Summarize the difference between an interacting and non-interacting Bose condensate. 11. Introduce the Bragg spectroscopy. 12. Show that the Bogoliubov theory fails in one-dimension due to the infrared divergence. 13. Discuss the basic idea of the Bethe-ansatz solution and the Tonks-Girardeau limit. 14. Introduce scale invariant quantum gases. 15. Introduce the Josephson effect and the self-trapping as two different transport phenomena driven by phase coherence. 16. Discuss the conjugate relation between the relative phase fluctuation and the relative particle number fluctuation, and show strong enough phase fluctuation renders the system into a Fock state. 17. Discuss an example of quantum measurement that can project two independent condensates to a coherent superposition state. 18. Discuss interference effects in density and density-density correlations. 19. Discuss the relation between the instability of the Sch\"odinger Cat state and the stability of symmetry breaking.

Cambridge University Press eBooksJan 30, 20210 citations

Bose-Einstein condensation in chains with long-reach couplings

Delvis Bertrand Jorge de Barros

In this work, we investigate the critical properties of the Bose-Einstein condensation transition in a model of non-interacting bosons whose one-particle hamiltonian in the tight-binding approximation includes long-range couplings in a linear chain with periodic boundary conditions. The couplings along the chain are described by a kinetic hopping term which is responsible by the quantum mobility of the particles and whose amplitude decays with the distance between the sites as a power-law with a characteristic exponent a. We show that the density of states in the vicinity of the lower band edge of the energy spectrum strongly depends on a. In the re- gime of 1 < α < 2, the density of states vanishes as a power-law, similar to the behavior found for bosons in hypercubic lattices with dimension d = 2/(α -1). We determine the critical properties of the Bose-Einstein condensation transition that takes place in this regime using a hypothesis of a single characteristic length scale and a finite-size scaling analysis. In the regime of 3/2 < α < 2, the critical exponents are quite similar to those that characterize the transition in the corresponding hypercubic lattice. On the other hand, in the regime of 1 < α < 3/2, the exponents obtained from the single length scale hypothesis are quite distinct form those on the hypercubic lattice. This result indicates that the single length scale hypothesis can not correctly capture the critical exponents in the regime where the effective dimensionality of the model is above the upper critical dimension above which the transition has a mean-field critical behavior.

Feb 20, 20140 citations

The 1D interacting Bose gas in a hard wall box

Murray T. Batchelor, Xin Guan, N. Oelkers +1 more

We consider the integrable one-dimensional δ-function interacting Bose gas in a hard wall box which is exactly solved via the coordinate Bethe ansatz. The ground-state energy, including the surface energy, is derived from the Lieb– Liniger-type integral equations. The leading and correction terms are obtained in the weak and strong coupling regimes from both the discrete Bethe equations and the integral equations. This allows the investigation of both finite-size and boundary effects in the integrable model. We also study the Luttinger liquid behaviour by calculating Luttinger parameters and correlations. The hard wall boundary conditions are seen to have a strong effect on the ground-state energy and phase correlations in the weak coupling regime. Enhancement of the local two-body correlations is shown by application of the Hellmann–Feynman theorem. PACS numbers: 05.30.−d, 67.40.Db, 05.30.Jp (Some figures in this article are in colour only in the electronic version) 1.

Mar 20, 201033 citations

Novel phases in long-range interacting quantum many-body systems

Tanul Gupta

Phases novatrices dans les systèmes quantiques à plusieurs corps avec interactions à longue portée Cette thèse explore les propriétés de l’état fondamental des systèmes bosoniques 1D fortement corrélés avec un saut à longue portée, où la cohérence quantique et les phénomènes collectifs induisent des comportements distincts de ceux des systèmes à interactions de courte portée. En utilisant des méthodes numériques avancées comme la Monte Carlo à intégrale de chemin et l’algorithme du ver, elle analyse les transitions de localisation induites par le désordre et les interactions, du superfluide à l’état non-superfluide, avec un saut décroissant selon une loi de puissance. Une découverte clé est une transition de phase quantique continue et invariante d’échelle, s’écartant du scénario attendu de Berezinskii-Kosterlitz-Thouless (BKT). Les résultats remettent en question la théorie de bosonisation et les études numériques de moyenne échelle, identifiant la gamme où les interactions à longue portée dominent. Ces conclusions ont des implications expérimentales pour les plateformes réalisant des modèles XY avec couplages en loi de puissance, comme les réseaux d'atomes de Rydberg et les chaînes d'ions piégés.

Dec 10, 20240 citations