We give a comprehensive introduction into a diagrammatic method that allows for the evaluation of Gutzwiller wave functions in finite spatial dimensions. We discuss in detail some numerical schemes that turned out to be useful in the real-space evalu
ation of the diagrams. The method is applied to the problem of d-wave superconductivity in a two-dimensional single-band Hubbard model. Here, we discuss in particular the role of long-range contributions in our diagrammatic expansion. We further reconsider our previous analysis on the kinetic energy gain in the superconducting state.
A systematic diagrammatic expansion for Gutzwiller-wave functions (DE-GWF) proposed very recently is used for the description of superconducting (SC) ground state in the two-dimensional square-lattice $t$-$J$ model with the hopping electron amplitude
s $t$ (and $t$) between nearest (and next-nearest) neighbors. On the example of the SC state analysis we provide a detailed comparison of the method results with other approaches. Namely: (i) the truncated DE-GWF method reproduces the variational Monte Carlo (VMC) results; (ii) in the lowest (zeroth) order of the expansion the method can reproduce the analytical results of the standard Gutzwiller approximation (GA), as well as of the recently proposed grand-canonical Gutzwiller approximation (GCGA). We obtain important features of the SC state. First, the SC gap at the Fermi surface resembles a $d_{x^2-y^2}$-wave only for optimally- and overdoped system, being diminished in the antinodal regions for the underdoped case in a qualitative agreement with experiment. Corrections to the gap structure are shown to arise from the longer range of the real-space pairing. Second, the nodal Fermi velocity is almost constant as a function of doping and agrees semi-quantitatively with experimental results. Third, we compare the doping dependence of the gap magnitude with experimental data. Fourth, we analyze the $mathbf{k}$-space properties of the model: Fermi surface topology and effective dispersion. The DE-GWF method opens up new perspectives for studying strongly-correlated systems, as: (i) it works in the thermodynamic limit, (ii) is comparable in accuracy to VMC, and (iii) has numerical complexity comparable to GA (i.e., it provides the results much faster than the VMC approach).
In this work we analyze the variational problem emerging from the Gutzwiller approach to strongly correlated systems. This problem comprises the two main steps: evaluation and minimization of the ground state energy $W$ for the postulated Gutzwiller
Wave Function (GWF). We discuss the available methods for evaluating $W$, in particular the recently proposed diagrammatic expansion method. We compare the two existing approaches to minimize $W$: the standard approach based on the effective single-particle Hamiltonian (EH) and the so-called Statistically-consistent Gutzwiller Approximation (SGA). On the example of the superconducting phase analysis we show that these approaches lead to the same minimum as it should be. However, the calculations within the SGA method are easier to perform and the two approaches allow for a simple cross-check of the obtained results. Finally, we show two ways of solving the equations resulting from the variational procedure, as well as how to incorporate the condition for a fixed number of particles.
A systematic diagrammatic expansion for Gutzwiller-wave functions (DE-GWF) is formulated and used for the description of superconducting (SC) ground state in the two-dimensional Hubbard model with electron-transfer amplitudes t (and t) between neares
t (and next-nearest) neighbors. The method is numerically very efficient and allows for a detailed analysis of the phase diagram as a function of all relevant parameters (U, delta, t) and a determination of the kinetic-energy driven pairing region. SC states appear only for substantial interactions, U/t > 3, and for not too large hole doping, delta < 0.32 for t = 0.25 t; this upper critical doping value agrees well with experiment for the cuprate high-temperature superconductors. We also obtain other important features of the SC state: (i) the SC gap at the Fermi surface resembles $d_{x^2-y^2}$-wave only around the optimal doping and the corrections to this state are shown to arise from the longer range of the pairing; (ii) the nodal Fermi velocity is almost constant as a function of doping and agrees quantitatively with the experimental results; (iii) the SC transition is driven by the kinetic-energy lowering for low doping and strong interactions.
In this brief overview we discuss the principal features of real space pairing as expressed via corresponding low-energy (t-J or periodic Anderson-Kondo) effective Hamiltonian, as well as consider concrete properties of those unconventional supercond
uctors. We also rise the basic question of statistical consistency within the so-called renormalized mean-field theory. In particular, we provide the phase diagrams encompassing the stable magnetic and superconducting states. We interpret real space pairing as correlated motion of fermion pair coupled by short-range exchange interaction of magnitude J comparable to the particle renormalized band energy $sim tx$, where $x$ is the carrier number per site. We also discuss briefly the difference between the real-space and the paramagnon - mediated sources of superconductivity. The paper concentrates both on recent novel results obtained in our research group, as well as puts the theoretical concepts in a conceptual as well as historical perspective. No slave-bosons are required to formulate the present approach.
The coexistence of antiferromagnetism with superconductivity is studied theoretically within the t-J model with the Zeeman term included. The strong electron correlations are accounted for by means of the extended Gutzwiller projection method within
a statistically-consistent approach proposed recently. The phase diagram on the band filling - magnetic field plane is shown, and subsequently the system properties are analyzed for the fixed band filling n=0.97. In this regime, the results reflect principal qualitative features observed recently in selected heavy fermion systems. Namely, (i) with the increasing magnetic field the system evolves from coexisting antiferromagnetic-superconducting phase, through antiferromagnetic phase, towards polarized paramagnetic state, and (ii) the onset of superconducting order suppresses partly the staggered moment. The superconducting gap has both the spin-singlet and the staggered-triplet components, a direct consequence of a coexistence of the superconducting state with antiferromagnetism.
We study conductance spectroscopy of a two-dimensional junction between a normal metal and a strongly-correlated superconductor in an applied magnetic field in the Pauli limit. Depending on the field strength the superconductor is either in the Barde
en-Cooper-Schrieffer (BCS), or in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state of the Fulde-Ferrell (FF) type. The strong correlations are accounted for by means of the Gutzwiller method what leads naturally to the emergence of the spin-dependent masses (SDM) of quasiparticles when the system is spin-polarized. The case without strong correlations (with the spin-independent masses, SIM) is analyzed for comparison. We consider both the s-wave and the d-wave symmetries of the superconducting gap and concentrate on the parallel orientation of the Cooper pair momentum Q with respect to the junction interface. The junction conductance is presented for selected barrier strengths (i.e., in the contact, intermediate, and tunneling limits). The conductance spectra in the cases with and without strong correlations differ essentially. Our analysis provides thus an experimentally accessible test for the presence of strong-correlations in the superconducting state. Namely, correlations alter the distance between the conductance peaks (or related conductance features) for carriers with spin-up and spin-down. In the uncorrelated case, this distance is twice the Zeeman energy. In the correlated case, the corresponding distance is about 30-50% smaller, but other models may provide even stronger difference, depending on details of the system electronic structure. It turns out that the strong correlations manifest themselves most clearly in the case of the junction with the BCS, rather than the FFLO superconductor, what should make the experimental verification of the present results simpler.
We consider Andreev reflection in a two dimensional junction between a normal metal and a heavy fermion superconductor in the Fulde-Ferrell (FF) type of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. We assume s-wave symmetry of the superconducti
ng gap. The parameters of the superconductor: the gap magnitude, the chemical potential, and the Cooper pair center-of-mass momentum Q, are all determined self-consistently within a mean-field (BCS) scheme. The Cooper pair momentum Q is chosen as perpendicular to the junction interface. We calculate the junction conductance for a series of barrier strengths. In the case of incoming electron with spin sigma = 1 only for magnetic fields close to the upper critical field H_{c2}, we obtain the so-called Andreev window i.e. the energy interval in which the reflection probability is maximal, which in turn is indicated by a peak in the conductance. The last result differs with other non-self-consistent calculations existing in the literature.
