The local entanglement Ev of the one-dimensional Hubbard model is studied on the basis of its Bethe-ansatz solution. The relationship between the local entanglement and the on-site Coulomb interaction U is obtained. Our results show that Ev is an even analytic function of U at half-filling and it reaches a maximum at the critical point U = 0. The variation of the local entanglement with the filling factor shows that the ground state with maximal symmetry possesses maximal entanglement. The magnetic field makes the local entanglement to decrease and approach to zero at saturated magnetization. The on-site Coulomb interaction always suppresses the local entanglement. Quantum entanglement, as one of the most intriguing feature of quantum theory, has been a subject of much study in recent years, mostly because its nonlocal connotation[1] is regarded as a valuable resource in quantum communication and information processing [2,3]. For the application purpose, much of recent attentions have been focused on entanglement relevant to realistic systems. For example, several authors have investigated entanglement in spin systems [4,5,6,7,8,9] as well as indistinguishable-particle systems [10,11]. The work of Osterloh et al. [7] and Osborne and Nielsen[8] on the XY model suggestively showing that the entanglement of two neighboring sites displays a sharp peak either near or at the critical point where quantum phase transition undergoes. Investigating the critical entanglement between a block of continuous spins and their supplemental parts in a spin chain model, Vidal et al. [9] pointed out its relation to the entropy in conformal field theories. Recently, we studied entanglement and quantum phase transition of the XXZ model [12] and obtained its dependences on the anisotropy parameter ∆ and correlation length ξ.It is well known that the Hubbard model is a typical model describing correlated Fermion systems. It plays a crucial role for understanding many physical phenomena in condensed matter physics, such as magnetic ordering, Mott-insulator transition, and superconductivity, etc. Therefore, the study of the entanglement in the Hubbard model not only provides possible clues for experimental realization, but also sheds new light on the understanding of quantum many-body systems.In this Letter, we study the local entanglement of onedimensional Hubbard model [13]. We obtain the dependence of the local entanglement on Coulomb interaction, particle number and magnetization respectively. We find that the ground state with maximal symmetry possesses the maximal local entanglement, and the Coulomb interaction always suppresses the local entanglement. To the best of our knowledge, no one has investigated the entanglement in the interacting many-fermion systems. Our results, which are based on the exact solution of the Hubbard model, will be inspirable for people to explore quantum entanglement and phase transition via nonperturbative approach for other interacting many-fermion systems. An important observation of our studies is tha...

We present a time-reversal invariant s-wave superconductor supporting Majorana edge modes. The multiband character of the model together with spin-orbit coupling are key to realizing such a topological superconductor. We characterize the topological phase diagram by using a partial Chern number sum, and show that the latter is physically related to the parity of the fermion number of the time-reversal invariant modes. By taking the self-consistency constraint on the s-wave pairing gap into account, we also establish the possibility of a direct topological superconductor-to-topological insulator quantum phase transition.

We investigate the emergence of universal dynamical scaling in quantum critical spin systems adiabatically driven out of equilibrium, with emphasis on quench dynamics which involves non-isolated critical points (i.e., critical regions) and cannot be a priori described through standard scaling arguments nor time-dependent perturbative approaches. Comparing to the case of an isolated quantum critical point, we find that non-equilibrium scaling behavior of a large class of physical observables may still be explained in terms of equilibrium critical exponents. However, the latter are in general non-trivially path-dependent, and detailed knowledge about the time-dependent excitation process becomes essential. In particular, we show how multiple level crossings within a gapless phase may completely suppress excitation depending on the control path. Our results typify non-ergodic scaling in continuous finite-order quantum phase transitions.

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We explore the robustness of universal dynamical scaling behavior in a quantum system near criticality with respect to initialization in a large class of states with finite energy. By focusing on a homogeneous XY quantum spin chain in a transverse field, we characterize the non-equilibrium response under adiabatic and sudden quench processes originating from a pure as well as a mixed excited initial state, and involving either a regular quantum critical or a multicritical point. We find that the critical exponents of the ground-state quantum phase transition can be encoded in the dynamical scaling exponents despite the finite energy of the initial state. In particular, we identify conditions on the initial distribution of quasi-particle excitation which ensure Kibble-Zurek scaling to persist. The emergence of effective thermal equilibrium behavior following a sudden quench towards criticality is also investigated, with focus on the long-time expectation value of the quasi-particle number operator. Despite the integrability of the XY model, this observable is found to behave thermally in quenches to a regular quantum critical point, provided that the system is initially prepared at sufficiently high temperature. However, a similar thermalization behavior fails to occur in quenches towards a multi-critical point. We argue that the observed lack of thermalization originates in this case in the asymmetry of the impulse region that is also responsible for anomalous multicritical dynamical scaling.