We present a detailed derivation of a multisite mean-field theory (MSMFT) used to describe the Mott-insulator to superfluid transition of bosonic atoms in optical lattices. The approach is based on partitioning the lattice into small clusters which are decoupled by means of a mean-field approximation. This approximation invokes local superfluid order parameters defined for each of the boundary sites of the cluster. The resulting MSMFT grand potential has a nontrivial topology as a function of the various order parameters. An understanding of this topology provides two different criteria for the determination of the Mott insulator superfluid phase boundaries. We apply this formalism to d-dimensional hypercubic lattices in one, two, and three dimensions and demonstrate the improvement in the estimation of the phase boundaries when MSMFT is utilized for increasingly larger clusters, with the best quantitative agreement found for d = 3. The MSMFT is then used to examine a linear dimer chain in which the onsite energies within the dimer have an energy separation of . This system has a complicated phase diagram within the parameter space of the model, with many distinct Mott phases separated by superfluid regions.MCINTOSH, PISARSKI, GOODING, AND ZAREMBA PHYSICAL REVIEW A 86, 013623 (2012) II. FORMALISM: MULTISITE MEAN-FIELD THEORY A. Derivation in one dimensionOur work is based on the BH Hamiltonian [2] in the grand canonical ensemble. With the assumption of a single orbital per site, this Hamiltonian is given bywhere the index i labels the sites of the optical lattice andĉ † i andĉ i are site creation and annihilation operators (henceforth referred to as site operators), respectively; the number operator for site i is given byn i =ĉ † iĉ i . The system parameters include the onsite energies ε i at each lattice site, the tunneling energy J ij between sites i and j , and the intrasite interaction energy U i . To a good approximation it is sufficient to ignore interactions between bosons on different sites and hopping between sites further apart than the nearest-neighbor distance [2,37]. Furthermore, we will restrict our considerations to the case where the interaction parameter has a common value U for all sites and a hopping parameter J for all nearest-neighbor pairs. Generalizations to more complex situations such as superlattices [15][16][17] can be readily accommodated in the MSMFT that we develop. The final parameter in the BH Hamiltonian is the chemical potential μ which controls the 013623-2 013623-3 MCINTOSH, PISARSKI, GOODING, AND ZAREMBA PHYSICAL REVIEW A 86, 013623 (2012)

We systematically develop a density functional description for the equilibrium properties of a two-dimensional, harmonically trapped, spin-polarized dipolar Fermi gas based on the Thomas-Fermi von Weizsäcker approximation. We pay particular attention to the construction of the twodimensional kinetic energy functional, where corrections beyond the local density approximation must be motivated with care. We also present an intuitive derivation of the interaction energy functional associated with the dipolar interactions, and provide physical insight into why it can be represented as a local functional. Finally, a simple, and highly efficient self-consistent numerical procedure is developed to determine the equilibrium density of the system for a range of dipole interaction strengths. PACS numbers: 31.15.E-, 71.10.Ca, 03.75.Ss, 05.30.Fk I. INTRODUCTIONUltra-cold, trapped dipolar quantum gases have received increasing attention over the past decade owing to the inherently interesting properties of the anisotropic, and long-range nature of the dipole-dipole interaction [1]. One of the important consequences of the anisotropy is that the interactions between the particles can be tuned from being predominantly attractive to repulsive by simply changing the 3D trapping geometry, or for dipoles confined to the 2D x-y plane, by adjusting the orientation of the dipoles relative to the z-axis [1, 2]. Therefore, novel physics in both the equilibrium and dynamic properties of such systems may be explored as a function of the strength of the interaction, the geometry of the confining potential, and the dimensionality of the system.While the degenerate dipolar Bose gas has been well studied experimentally and theoretically [1], realizing a degenerate dipolar Fermi gas in the laboratory has proven to be much more elusive. One of the reasons for this is that the path to quantum degeneracy is impeded by the Pauli principle, which forbids s-wave collisions between identical atoms. Thus, early attempts to cool both magnetic and molecular dipolar Fermi gases below degeneracy were unsuccessful [3][4][5][6]. However, in the recent work of M. Lu et al. [7], this experimental hurdle was finally overcome, resulting in the experimental realization of a spin polarized, degenerate dipolar Fermi gas. Specifically, using the method of sympathetic cooling, a mixture consisting of 161 Dy and the bosonic isotope 162 Dy, were cooled to T /T F ∼ 0.2. In addition, this group was also able to evaporatively cool a single component gas of 161 Dy down to a temperature of T /T F ∼ 0.7. This latter result is presumed to arise from the rethermalization provided by the strong dipolar scattering between the 161 Dy atoms which have a large magnetic moment (µ ∼ 10µ B ).The ability to fabricate such systems in the laboratory now opens the door for the investigation of both the equilibrium and dynamical properties of dipolar Fermi gases, and will enable contact to be made with the large body of theoretical work already in the literature [1]. Moreover, it is now r...

We present a multisite formulation of mean-field theory applied to the disordered Bose-Hubbard model. In this approach the lattice is partitioned into clusters, each isolated cluster being treated exactly, with intercluster hopping being treated approximately. The theory allows for the possibility of a different superfluid order parameter at every site in the lattice, such as what has been used in previously published site-decoupled mean-field theories, but a multisite formulation also allows for the inclusion of spatial correlations allowing us, e.g., to calculate the correlation length (over the length scale of each cluster). We present our numerical results for a two-dimensional system. This theory is shown to produce a phase diagram in which the stability of the Mott-insulator phase is larger than that predicted by site-decoupled single-site mean-field theory. Two different methods are given for the identification of the Bose-glass-to-superfluid transition, one an approximation based on the behavior of the condensate fraction, and one that relies on obtaining the spatial variation of the order parameter correlation. The relation of our results to a recent proposal that both transitions are non-self-averaging is discussed.

The average-density approximation is used to construct a nonlocal kinetic energy functional for an inhomogeneous two-dimensional Fermi gas. This functional is then used to formulate a ThomasFermi von Weizsäcker-like theory for the description of the ground state properties of the system. The quality of the kinetic energy functional is tested by performing a fully self-consistent calculation for an ideal, harmonically confined, two-dimensional system. Good agreement with exact results are found, with the number and kinetic energy densities exhibiting oscillatory structure associated with the nonlocality of the energy functional. Most importantly, this functional shows a marked improvement over the two-dimensional Thomas-Fermi von Weizsäcker theory, particularly in the vicinity of the classically forbidden region.

We study a single impurity effect on the local density of states in a d-wave superconductor accounting for the momentum-dependent impurity potential. We show that the anisotropy of the scattering potential can alter significantly the spatial dependence of the quasiparticle density of states in the vicinity of the impurity.A search for the symmetry of the cuprate superconducting state involves among others theoretical studies of the disordered systems [1,2,3]. The most direct probe of a simple defect impact on superconductivity is provided by the scanning tunneling microscopy (STM) measurement of the position-dependent quasiparticle density of states. The STM images of Zn and Ni substitutions at the planar Cu sites in Bi 2 Sr 2 CaCu 2 O 8+δ reveal a distinct four-fold symmetry of the local density of states (LDOS) [4,5] predicted for the d-wave superconductor response to disorder [6,7]. In addition to providing detailed information on the superconducting state, this kind of experiment may shed light on the nature of the quasiparticle scattering centers. Although the main feature of the four-fold symmetry of the tunneling currents is captured by the model of isotropic impurity scattering the observed spatial dependence of the quasiparticle density of states is far more complex, and may originate from a non trivial structure of the impurity potential. This aspect of the quasiparticle scattering process has been raised in the discussion of macroscopic measurements in these compounds [8,9,10,11]. It has been shown that the unexpected for the d-wave superconductivity weak impurity-induced suppression of the critical temperature can be understood within a scenario of a momentum-dependent (anisotropic) impurity scattering potential [12,13,14,15]. In the present paper we discuss the effect of anisotropic impurity potential on the local density of states in the vicinity of a single impurity and verify to what extend this scenario can reproduce the STM maps of Bi 2 Sr 2 CaCu 2 O 8+δ . There are studies of the momentum-dependent impurity scattering which indicate the existence of the impurity-bound states even for the potential strength far from resonant [17]. Therefore, we focus here on the symmetry of the LDOS and its spatial dependence. For this purpose and for analytical simplicity we consider scattering in the Born approximation. Hereafter we show that the anisotropy of the scattering potential alters the local density of states in the d-wave superconductor but preserves its tetragonal symmetry. Compared to the effect of the isotropic impurity potential it enhances the spatial variation of the quasiparticle density of states and can lead to a longrange spatial modulation of the LDOS spectra. We note that the anisotropy of the impurity potential, if the same as the anisotropy of the superconducting order parameter, enhances the relative magnitude of oscillations in the LDOS for the direction of a gap maximum, especially in the vicinity of the impurity, whereas the impurity potential with maxima in the nodal region inver...

Fourier transformed maps of the local density of states in disordered superconducting 2 2 2 8Bi Sr CaCu O d + provided important information on the quasiparticle states and their interference in high-temperature superconductors. Neglecting the momentum-dependence of the impurity potential it is possible to disentangle the quasiparticle interference effect from the static structure factor of the scatterers. The k-dependence of the impurity potential makes such a distinction difficult. Here, for the model momentum-dependent potential, we discuss the quasiparticle interference induced by an impurity in an s-wave superconductor. . However, despite providing a detailed information on the superconducting state [4,5], the local density of states (LDOS) obtained in the STM measurement represents a response of the system to the impurity potential and must be regarded as a joint effect of the quasiparticle spectrum and the perturbation potential. We have already shown that the momentum-dependent, or extended in the real space, impurity potential alters the LDOS of tetragonal superconductors [6,7]. A new method of studying the quasiparticle density of states has been provided by a high-resolution Fourier-transform scanning-tunneling microscopy (FT-STM) [8,9]. It has been shown that the effects of the quasiparticle interference and of the scattering potential structure can be disentangled in the analysis of the Fouriertransformed LDOS upon the assumption of the on-site impurity potential. Such an assumption allowed the interpretation of the FT-STM maps as a quasiparticle interference picture determined by the characteristic wave vectors connecting constant energy regions of the quasiparticle excitations in a superconducting state [10,11].In this paper we reconsider this assumption for an isotropic two-dimensional s-wave superconductor and study the effect of quasiparticle interference induced by the momentum-dependent impurity potential. We show that such a scattering process leads to anisotropic interference patterns resembling the ones associated with a non trivial symmetry of the superconducting ground state. Our calculations are done in Nambu notation, ˆi t ( 1 2 3 i = , , ) are the Pauli matrices and 0 t is the identity matrix in the particle-hole space.We consider the effect of a weak scattering by the momentum-dependent impurity potential

We discuss the effect of an anisotropic impurity potential on the critical temperature, local density of states in the vicinity of a single impurity, and the quasiparticle density of states for a finite impurity concentration in a d-wave superconductor. Different scattering regimes are concerned.

Quantum interference revealed in Bi 2 Sr 2 CaCu 2 O 8+δ by Fourier transform scanning spectroscopy is primarily determined by maxima of the joint density of states which correspond to an octet of scattering wave vectors connecting points at the constant quasiparticle energy contours where the density of states is maximal. We consider a superconductor with a constant quasiparticle density of states where the interference is due to the fourfold change in the phase of the order parameter. Our results may be helpful in a discussion of the interference in a d-wave superconductor at the tunneling energy exceeding the gap magnitude.