We present an approach to the Bose-Hubbard model using the U(1) quantum rotor description. The effective action formalism allows us to formulate a problem in the phase only action and obtain analytical formulas for the critical lines. We show that the nontrivial U(1) phase field configurations have an impact on the phase diagrams. The topological character of the quantum field is governed by terms of the integer charges—winding numbers. The comparison of presented results to recently obtained quantum Monte Carlo numerical calculations suggests that the competition between quantum effects in strongly interacting boson systems is correctly captured by our model.

The infinite-range quantum Ising spin glass in transverse ($\Gamma$) and longitudinal ($h$) fields is studied by means of thermo-field-dynamics as a substitute for the $n$-replica trick. With the use of a one-loop approximation for the dynamic self-interaction the stability of the mean-field-type solution against the action of fluctuations is investigated in the parameter space ($k_{B}T, \Gamma , h$).

The electronic $t\text{−}{t}^{\prime }\text{−}U\text{−}J$ model for the $\mathrm{Cu}{\mathrm{O}}_2$ plane in high-$T_c$ superconducting cuprates is elaborated. It incorporates both charge transfer $U$ and exchange correlation $J$ with direct $t$ and diagonal $t^{\prime }$ hopping. Making use of the Yang concept of the off diagonal long range order (ODLRO) in the reduced density matrix of the electronic system we study the loss of quantum wave coherence (and thereby superconductivity) due to the Coulomb interaction. To this aim the second-quantized Hamiltonian of the model is translated to the phase representation with the help of the topologically constrained path integral formalism. As a result the electrons emerge as composite particles consisting of of spin-carrying neutral fermions and charged topological bosons in a form of “flux tubes” with the quantum phase variable dual to the local electron density. The charge carrying bose field forms a collective variable representing the correlated “sliding” motion of the entire electronic system. We found that the spin gap $\Delta$ which appears in one-particle spectrum is the result of valence bond pairing of neutral fermions without ODLRO and deduce the solution for $\Delta$ with “$d$-wave” symmetry. In the bosonic sector we derive the effective quantum rotor model along with microscopic phase stiffnesses setting the energy scale for superconductivity and point out the special role played by the diagonal hopping $t^{\prime }$. We explore the relationship of the quantum rotor phase model and the Bose-Hubbard system emphasizing the ring topology of quantum phase variable. Furthermore, we obtain the ground state phase diagrams which facilitate the Mott insulator-superconductor transition scenario. We elucidate the role of the chemical potential (doping) and the huge degeneracy of the effective bosonic system in achieving ODLRO with long-range phase coherence in the presence of large Coulomb repulsion. Finally, we discuss a number of relevant issues pertaining to the the present work which include: theoretical aspects of the hierarchy of the energy scales in cuprates, “spin-charge separation” concepts, “local-pair” and “phase-fluctuation” scenarios as well as experimental findings related to the role of the “$d$-wave” symmetry of the spin gap and diagonal hopping $t^{\prime }$ in cuprates.

We have studied a quantum Hamiltonian that models an array of ultrasmall Josephson junctions with short-range Josephson couplings $E_J$ and charging energies $E_C$ due to the small capacitance of the junctions. We derive a new effective quantum spherical model for the array Hamiltonian. As an application we start by approximating the capacitance matrix by its self-capacitive limit and in the presence of an external uniform background of charges $q_x.$ In this limit we obtain the zero-temperature superconductor-insulator phase diagram $E_J^{\mathrm{crit}}{(E}_C{,q}_x)$ that improves upon previous theoretical results that used a mean-field theory approximation. Next we obtain a closed-form expression for the conductivity of a square array, and derive a universal scaling relation valid about the zero-temperature quantum critical point. In the latter regime the energy scale is determined by temperature and we establish universal scaling forms for the frequency dependence of the conductivity.

We study the spatial correlations of strongly interacting bosons in a ground state, confined in a two-dimensional square and a three-dimensional cubic lattice. Using the combined Bogoliubov method and the quantum rotor approach, we map the Hamiltonian of strongly interacting bosons onto U(1) phase action in order to calculate the atom-atom correlations’ decay along the principal axis and a diagonal of the lattice-plane direction as a function of distance. Lower tunneling rates lead to quicker decays of the correlations, whose character becomes exponential. Finally, correlation functions allow us to calculate quantities that are directly bound to experimental outcomes, namely time-of-flight absorption images and resulting visibility. Our results contain all the characteristic features present in experimental data (transition from Mott insulating blob to superfluid peaks, etc.), emphasizing the usability of the proposed approach. DOI: http://dx.doi.org/10.1103/PhysRevA.84.053613 ©2011 American Physical Society