Localization properties of non-interacting quantum particles in one-dimensional incommensurate lattices are investigated with an exponential short-range hopping that is beyond the minimal nearestneighbor tight-binding model. Energy dependent mobility edges are analytically predicted in this model and verified with numerical calculations. The results are then mapped to the continuum Schrödinger equation, and an approximate analytical expression for the localization phase diagram and the energy dependent mobility edges in the ground band is obtained.PACS numbers: 37.10.Jk; 05.60.Gg Anderson localization, the localization of electronic Bloch waves due to interference in disordered potentials, is one of the fundamental quantum phenomena in nature and is the transport mechanism behind metalinsulator phase transitions in solids [1]. Although this mechanism was first proposed over 50 years ago, direct observation of Anderson localization has been notoriously difficult due to the problems in reliably controlling disorder in solid-state systems. But recent advances in the manipulation of ultra-cold atoms offer a completely new, well-controlled tool in directly observing such fundamental quantum phenomena. A notable example is the recent work done by Billy et. al. who observed Anderson localization in a diffuse Bose-Einstein condensate in a 1 dimensional (1D) waveguide with a disordered laser speckle potential [2]. Another recent example is the work done by Roati et. al. who observed 1D Aubry-André localization (a phase transition closely related to Anderson localization[3]) of cold-atoms in an incommensurate quasi-periodic potential [4]. These advances highlight the potential of ultra-cold atoms to experimentally probe fundamental quantum localization phenomena that previously could only be studied indirectly or through numerical calculations. Cold atomic systems offer precise control of the background potential, and the non-interacting limit is easily achievable with a very dilute gas of either bosons or fermions. This is the context (and the motivation) of the current work, where we introduce a new and theoretically exact 1D localization model with mobility edges that should be observable in cold atomic systems.1D localization phenomena are traditionally studied with the nearest neighbor tight binding model:where t 1 is the hopping term representing tunneling between nearest neighboring sites and V n is the onsite disordered potential [1,2] or the incommensurate potential [3,4]. The simplicity of (1) [6,7]. Ultra-cold atoms loaded into optical lattices with controllable depths provide an experimental tool to study transport beyond the tight binding regime, where mobility edges are likely. In this Letter, we introduce an exact analytically solvable 1D localization model which has a energy dependent mobility edge.We believe that our model should be realizable in ultracold 1D atomic systems, and we show that our theoretical findings extend to the general Schrödinger equation description well outside the tight binding ...

Water shows intriguing thermodynamic and dynamic anomalies in the supercooled liquid state. One possible explanation of the origin of these anomalies lies in the existence of a metastable liquid-liquid phase transition (LLPT) between two (high and low density) forms of water. While the anomalies are observed in experiments on bulk and confined water and by computer simulation studies of different water-like models, the existence of a LLPT in water is still debated. Unambiguous experimental proof of the existence of a LLPT in bulk supercooled water is hampered by fast ice nucleation which is a precursor of the hypothesized LLPT. Moreover, the hypothesized LLPT, being metastable, in principle cannot exist in the thermodynamic limit (infinite size, infinite time). Therefore, computer simulations of water models are crucial for exploring the possibility of the metastable LLPT and the nature of the anomalies. In this work, we present new simulation results in the NVT ensemble for one of the most accurate classical molecular models of water, TIP4P/2005. To describe the computed properties and explore the possibility of a LLPT, we have applied two-structure thermodynamics, viewing water as a non-ideal mixture of two interconvertible local structures ("states"). The results suggest the presence of a liquid-liquid critical point and are consistent with the existence of a LLPT in this model for the simulated length and time scales. We have compared the behavior of TIP4P/2005 with other popular water-like models, namely, mW and ST2, and with real water, all of which are well described by two-state thermodynamics. In view of the current debate involving different studies of TIP4P/2005, we discuss consequences of metastability and finite size in observing the liquid-liquid separation. We also address the relationship between the phenomenological order parameter of two-structure thermodynamics and the microscopic nature of the low-density structure.

Localization properties of particles in one-dimensional incommensurate lattices without interaction are investigated with models beyond the tight-binding Aubry-André (AA) model. Based on a tight-binding t1 − t2 model with finite next-nearest-neighbor hopping t2, we find the localization properties qualitatively different from those of the AA model, signaled by the appearance of mobility edges. We then further go beyond the tight-binding assumption and directly study the system based on the more fundamental single-particle Schrödinger equation. With this approach, we also observe the presence of mobility edges and localization properties dependent on incommensuration.PACS numbers: 37.10.Jk, The physics of quantum transport in random disordered potentials has been a subject of substantial interest for condensed matter physicists for decades. The extended Bloch waves in a periodic lattice could undergo a quantum interference induced transition into localized states due to random disorder by a mechanism commonly referred to as Anderson localization [1]. Matter waves can also be localized in deterministic potentials that exhibit some similarities to random disorder [2,3,4,5]. Quasi-periodic potentials, such as incommensurate lattices (the superposition of two or more lattices with incommensurate periods), are notable examples and have been extensively studied with the Aubry-André model [2]. Such potentials have been shown to exhibit interesting quantum transport phenomena in themselves. Incommensurate potentials, for example, are theorized to have fractal spectrums [6]. However, it remains challenging to study these phenomena in solid state experiments, as it is difficult to systematically control the disorder in solid state systems. In contrast to the solid state systems, ultracold atoms loaded in optical lattices offer remarkable controllability over the system parameters, making it an attractive platform for the study of the localization of matter waves. Recently, Anderson localization of noninteracting Bose-Einstein condensates (BEC) has been observed in a one-dimensional matter waveguide with a random potential introduced with laser speckles [7]. Similar experiments have also been done in quasi-periodic optical lattices [8,9].Localization of noninteracting particles in one dimensional incommensurate lattices is often studied with the Aubry-André model (AA) with nearest neighbor (nn) hopping, where one of the lattices is assumed to be relatively weak and can be treated as a perturbation . Within the framework of the AA model, there is a duality point, at which a sharp transition from all eigenstates being extended to all being localized occurs. However, in ultracold atom experiments, one can tune the depth of each lattice in a controllable way and bring the system out of the tight-binding regime. To explore the physics of localization for shallow lattices, it is of interest to go beyond the AA model and the tight-binding assumption [10].In this work, we first study the tight-binding t 1 − t 2 model, which extends...

We study the quantum localization phenomena of noninteracting particles in one-dimensional lattices based on tight-binding models with various forms of hopping terms beyond the nearest neighbor, which are generalizations of the famous Aubry-André and noninteracting Anderson model. For the case with deterministic disordered potential induced by a secondary incommensurate lattice (i.e. the Aubry-André model), we identify a class of self-dual models, for which the boundary between localized and extended eigenstates are determined analytically by employing a generalized Aubry-André transformation. We also numerically investigate the localization properties of nondual models with next-nearest-neighbor hopping, Gaussian, and power-law decay hopping terms. We find that even for these non-dual models, the numerically obtained mobility edges can be well approximated by the analytically obtained condition for localization transition in the self dual models, as long as the decay of the hopping rate with respect to distance is sufficiently fast. For the disordered potential with genuinely random character, we examine scenarios with next-nearestneighbor hopping, exponential, Gaussian, and power-law decay hopping terms numerically. We find that the higher-order hopping terms can remove the symmetry in the localization length about the energy band center compared to the Anderson model. Furthermore, our results demonstrate that for the power-law decay case, there exists a critical exponent below which mobility edges can be found. Our theoretical results could, in principle, be directly tested in shallow atomic optical lattice systems enabling non-nearest-neighbor hopping.

One of the most promising frameworks for understanding the anomalies of cold and supercooled water postulates the existence of two competing, interconvertible local structures. If the non-ideality in the Gibbs energy of mixing overcomes the ideal entropy of mixing of these two structures, a liquid-liquid phase transition, terminated at a liquid-liquid critical point, is predicted. Various versions of the "twostructure equation of state" (TSEOS) based on this concept have shown remarkable agreement with both experimental data for metastable, deeply supercooled water and simulations of molecular water models. However, existing TSEOSs were not designed to describe the negative pressure region and do not account for the stability limit of the liquid state with respect to the vapor. While experimental data on supercooled water at negative pressures may shed additional light on the source of the anomalies of water, such data are very limited. To fill this gap, we have analyzed simulation results for TIP4P/2005, one of the most accurate classical water models available. We have used recently published simulation data, and performed additional simulations, over a broad range of positive and negative pressures, from ambient temperature to deeply supercooled conditions. We show that, by explicitly incorporating the liquid-vapor spinodal into a TSEOS, we are able to match the simulation data for TIP4P/2005 with remarkable accuracy. In particular, this equation of state quantitatively reproduces the lines of extrema in density, isothermal compressibility, and isobaric heat capacity. Contrary to an explanation of the thermodynamic anomalies of water based on a "retracing spinodal," the liquid-vapor spinodal in the present TSEOS continues monotonically to lower pressures upon cooling, influencing but not giving rise to density extrema and other thermodynamic anomalies. Published by AIP Publishing.