Session B23: Superconductivity Theory I: Cluster DMFT, t-J model, renormalization group

Pair Structure and the Pairing Interaction in a Bilayer Hubbard model

The bilayer Hubbard model with an intra-layer hopping $t$ and an inter-layer hopping $t_\perp$ provides an interesting testing ground for several aspects of what has been called unconventional superconductivity. One can study the type of pair structures which arise when there are multiple Fermi surfaces. One can also examine the pairing for a system in which the structure of the spin-fluctuation spectral weight can be changed. Using a dynamic cluster quantum Monte Carlo approximation, we find that near half-filling, if the splitting between the bonding and anti-bonding bands $t_\perp/t$ is small, the gap has $B_{1g}$ ($d_{x^2-y^2}$-wave) symmetry but when the splitting becomes larger, $A_{1g}$ ($s^\pm$-wave) pairing is favored. We also find that in the $s^\pm$ pairing region, the pairing is driven by inter-layer spin fluctuations and that $T_c$ is enhanced.   

Influence of the cluster-shape on the d-wave-transition temperature $T_c$ in the dynamical cluster approximation: A case study for the 2D repulsive Hubbard model

The dynamical cluster approximation (DCA) is a systematic extension beyond the single site approximation of dynamical mean field theory to include non-local correlations. The method maps the infinite lattice self-consistently on a cluster with $N_c$ sites, which is embedded into a mean field. Since the correlations within the cluster are treated exactly, it is ideal to investigate phase-transitions such as d-wave superconductivity. Since a Quantum Monte Carlo integration is used to obtain the DCA self-consistency, the sign-problem prevents us to solve large cluster problems ($N_c>32$). Meanwhile, the transition temperatures $T_c$ from smaller clusters fluctuate due to a large dependency on the cluster-shape. Here, we investigate whether this cluster-dependency originates from the quantum impurity problem, or from the mesh on which we solve the Bethe-Salpeter (BS) equation. By exploiting the localized nature of the self-energy $\Sigma$ and the vertex function $\Gamma$ in real space, we can reconstruct these functions on arbitrary fine meshes in momentum-space and investigate its impact on the solutions of the BS equation. We will show that the $8$-site cluster has a finite $T_c$ for small doping-levels and that the 16-site clusters have converged transition temperatures.   

Superconductivity and the Pseudogap in the 2d Hubbard model

Using a numerically exact continuous-time quantum Monte Carlo impurity solver and the DCA cluster dynamical mean field method with cluster sizes up to 16, we have been able to access the superconducting phase of the two dimensional Hubbard model for parameters believed to be relevant to high temperature copper oxide superconductivity. We present results for the phase diagram, the gap to transition temperature ratio, and the interplay of the pseudogap and the superconducting gap. The gap results are obtained by direct inference from imaginary frequency data and analytically continued spectral functions.   

Strongly correlated superconductivity and Mott transition

Whether the pseudogap temperature $T^*$ intercepts or merges with the superconducting dome is one of the key questions in the field of high-temperature superconductors. We study the normal and the d-wave superconducting phases at finite temperature in the two-dimensional Hubbard model within cellular dynamical mean-field theory and continuous-time quantum Monte Carlo. Above the critical value for the Mott transition, the superconducting $T_c$ has a dome-like shape as a function of doping. The pseudogap temperature $T^*$ intercepts the superconducting dome. Removing superconductivity, one finds that in the normal state, $T^*$ ends at a finite-doping first-order transition that occurs at temperatures below the superconducting dome. That first order transition between a pseudogap metal and a strongly correlated metal is linked to the Mott transition at half-filling. Refs: G. Sordi et al., PRL 104, 226402 (2010); G. Sordi et al., PRB 84, 075161 (2011); G. Sordi et al., arXiv:1110.1392 (2011).   

Multi-band Hubbard Model Simulation for Unconventional Superconducting System

To simulate the new experiment findings on unconventional superconductor materials such as Cuprates and iron pnictides, it is inevitable to go beyond single-band model and capture the physics from multi-band effects. We carried out a novel Dynamical Cluster Quantum Monte Carlo study on multi-band Hubbard model, implementing the Continuous Time Quantum Monte Carlo algorithm as quantum solver. We studied various of single-particle properties aiming to observe the experiment concerned issues such as the spontaneous symmetry breaking to nematic order.   

Scaling of the transition temperature of hole-doped cuprate superconductors with the charge-transfer energy

We use first-principles calculations to extract two essential microscopic parameters, the charge-transfer energy and the inter-cell oxygen-oxygen hopping, which correlate with the maximum superconducting transition temperature $Tcmax$ across the cuprates. We explore the superconducting state in the three-band model of the copper-oxygen planes using cluster Dynamical Mean-Field Theory with an exact diagonalization impurity solver. We find that variations in the charge-transfer energy largely accounts for the trend in $Tcmax$ across the cuprate families.   

A contractor-renormalization study of Hubbard plaquette clusters

We implement the contractor-renormalization method to study the checkerboard Hubbard model on various finite-size clusters as function of the inter-plaquette hopping $t'$ and the on-site repulsion $U$ at low hole doping. We find that the pair-binding energy and the spin gap exhibit a pronounced maximum at intermediate values of $t'$ and $U$, thus indicating that moderate inhomogeneity of the type considered here substantially enhances the formation of hole pairs. The rise of the pair-binding energy for $t'< t'_{\rm max}$ is kinetic-energy driven and reflects the strong resonating valence bond correlations in the ground state that facilitate the motion of bound pairs as compared to single holes. Conversely, as $t'$ is increased beyond $t'_{\rm max}$ antiferromagnetic magnons proliferate and reduce the potential energy of unpaired holes and with it the pairing strength. For the periodic clusters that we study the estimated phase ordering temperature at $t'=t'_{\rm max}$ is a factor of 2--6 smaller than the pairing temperature.   

Effects of disorder in the Checkerboard Hubbard model

The checkerboard Hubbard model (CHM) is an unusual example of a model for strongly correlated electrons that has a region in parameter space where a controlled perturbative solution is possible. The square lattice in two dimensions is divided into four-site plaquettes for which the intra-plaquette hopping ($t$) is stronger than the inter-plaquette hopping ($t'$). We study the ground state properties of the CHM in the presence of disorder using exact diagonalization on clusters of up to twelve sites as a function of $t'/t$, disorder strength and interaction strength. We consider both site and bond disorder and calculate the pair binding energy and the spin gap. We comment on the implications of our results for superconductivity in this model.   

Superconducting pair-pair correlations in the half-filled Hubbard model on the anisotropic triangular lattice: absence of long-range order

We report calculations of superconducting pair-pair correlations for the half-filled band Hubbard model on an anisotropic triangular lattice using the path-integral renormalization group (PIRG) method. Mean-field studies have suggested that $d_{x^2-y^2}$ superconductivity occurs near the boundary between metallic and antiferromagnetic phases in this model. We calculate bond orders, spin structure factors, and superconducting pair-pair correlations at zero temperature. Our results are consistent with previous studies of the metal-insulator transition and antiferromagnetism in this model. However, we do not find any parameter region where pair-pair correlations are enhanced by the Hubbard U, except for trivial enhancement of on-site correlations. The superconducting pair-pair correlations at larger distances decrease monotonically with increasing U, with a distance dependence approaching that of noninteracting fermions, indicating the absence of frustration-driven superconductivity within the model.   

Two temperature scales in the d-wave pair correlation length of the 2D t-J model

Two temperature scales are observed in the d-wave pair correlation length for doping $\delta\sim0.25$. Both the s-wave and d-wave pair correlation lengths increase with decreasing temperature starting at very high temperatures. By $T\sim3J$ both symmetries have grown to $\sim1/3$ of a lattice spacing. For $T>3J$ the s-wave correlation length is a few percent larger than d-wave. At $T\sim3J$ both symmetries are saturating, with the growth in the correlation length flattening for both symmetries. The s-wave correlation length does not grow at lower temperatures. The d-wave correlation length does resume growing at $T\sim0.8J$, with the growth at low temperatures much stronger than it was at high temperatures. The d-wave correlation length reaches a size of one lattice spacing at $T\sim0.25J$ with no sign of saturating. Interestingly, both of these temperature scales also occur in the t-J model momentum distribution $n_{\bf k}$. The features that are found in the temperature dependence of the 2D $n_{\bf k}$ on the zone diagonal are very similar to the temperature dependence of the 1D t-J model $n_{\bf k}$. These observations can be understood by having spin-charge separated degrees of freedom.   

Two routes to disorder-induced magnetism and nematicity in the cuprates

We study disorder-induced magnetism within the Gutzwiller approximation applied to the t-J model relevant for cuprate superconductors. We identify two distinct disorder-induced magnetic phases depending on the strength of the scatterers. For weak potential scatterers, charge reorganization may push local regions in-between the impurities across the magnetic phase boundary, whereas for strong scatterers a local static magnetic moment is formed around each impurity. We calculate the density of states and find a universal low-energy behavior independent of both disorder and magnetization. However, the magnetic regions are characterized by larger (reduced) superconducting gap (coherence peaks) [1]. Recent studies have highlighted the role of a electronic nematic liquid underdoped cuprates. We calculate the spin susceptibility with a small explicitly broken rotational symmetry to show how the induced spin response asymmetry is enhanced by correlations. In the disorder-induced stripe phase, impurities become spin nematogens with a C2 symmetric impurity resonance state, and the disorder-averaged spin susceptibility remains only C2 symmetric at low energies, similar to recent data from neutron scattering on underdoped YBCO [2].\\[4pt] [1] R. B. Christensen et al., accepted Phys. Rev. B (2011).\\[0pt] [2] B. M. Andersen et al., submitted to EPL (2011).   

Band structure effects on superconductivity in the weak-coupling Hubbard model

The repulsive Hubbard model is the paradigmatic model for the study of unconventional superconductivity. In order to explore the influence of various features of the band structure on the magnitude and character of the pairing, we use a well-controlled perturbative renormalization group (RG) method to study the weak coupling limit of the model on the square lattice with various modifications: The first is the checkerboard model described by the strong hopping $t$ and the weak hopping $t^{\prime}$. The second is the bilayer model described by the intra-layer hopping $t$ and the inter-layer hopping $t_{\perp}$. We obtain the pairing symmetry and strength as functions of the relevant band parameters. Large changes in the effective pairing strength are found and their origins explained.   

Effects of longer-range interactions on unconventional superconductivity

We analyze the effect of the non-vanishing range of electron-electron repulsion on the mechanism of unconventional superconductivity. We present asymptotically exact weak-coupling results for dilute electrons in the continuum and for the 2D extended Hubbard model, as well as density-matrix renormalization group results for the two-leg extended Hubbard model at intermediate couplings, and approximate results for the case of realistically screened Coulomb interactions. We show that $T_c$ is generally suppressed in some pairing channels as longer range interactions increase in strength, but superconductivity is not destroyed. Our results confirm that electron-electron interaction can lead to unconventional superconductivity under physically realistic circumstances.   

Three-particle interactions in effective one-band models for unconventional superconductivity

We discuss different approximations for effective low-energy interactions in multiband models for weakly interacting correlated electrons. In the study of Fermi surface instabilities of the conduction band(s), the standard approximation consists in keeping just those terms in the bare interactions that couple only to the conduction band(s), while corrections due to virtual excitations into bands away from the Fermi surface are typically neglected. In order to include important aspects of these virtual interband excitations, we present an improved truncation of the functional renormalization group (fRG) that keeps track of the three-particle vertex in the conduction band. Within a simplified two-patch treatment of a two-band model, we demonstrate that these corrections can have a rather strong effect in parts of the phase diagram by changing the critical scale for $d $-wave pairing close to a phase boundary. The improved truncation scheme is applied as well to the Emery model within a multi-patch channel-decomposed fRG.   

Effective interactions in multi-band systems from constrained summations

The application of many-body techniques for the study of correlation effects and unconventional superconductivity requires the formulation of an effective low-energy model that contains only the relevant bands near the Fermi level. However the bands away from the Fermi level are known to renormalize the low-energy interactions substantially. Here we compare different schemes to derive low-energy effective theories for interacting electrons in solids. The frequently used constrained random phase approximation (cRPA) is identified as a particular resummation of higher-order interaction terms that includes important parts of of the leading virtual corrections. We then propose an adapted renormalization group scheme that includes the cPRA, but also allows one to go beyond the cRPA approximation. We study a simple two-band model in order to demonstrate the differences between the different approximations.