Variational Cluster Approximation Research Articles

Theory of the waterfall phenomenon in cuprate superconductors

Based on exact diagonalization and variational cluster approximation calculations we study the relationship between charge transfer models and the corresponding single band Hubbard models. We present an explanation for the waterfall phenomenon observed in angle resolved photoemission spectroscopy on cuprate superconductors. The phenomenon is due to the destructive interference between the phases of the O2$p$ orbitals belonging to a given Zhang-Rice singlet and the Bloch phases of the photohole which occurs in certain regions of $\mathbf{k}$ space. It therefore may be viewed as a direct experimental visualization of the Zhang-Rice construction of an effective single band model for the CuO${}_{2}$ plane.

Variational cluster approximation study of the one-dimensional Holstein-Hubbard model at half filling

The half-filled one-dimensional Holstein-Hubbard model presents, at zero temperature, a charge-density-wave (CDW) phase and a Mott insulator phase. Recent results have shown that the transition from one phase to the other might proceed through an intermediate metallic phase. In this work, we determine the CDW phase boundary using the variational cluster approximation. Using exact diagonalization and cluster perturbation theory, we study both the pair susceptibility and the spectral gaps in the non-CDW part of the phase diagram. We cannot rule out the existence of an intermediate metallic phase.

Microscopic Study of Electronic and Magnetic Properties for Ir Oxide

The exact diagonalization and the variational cluster approximation (VCA) are used to study the nature of a novel Mott insulator induced by a strong spin-orbit coupling for a two-dimensional three-band Hubbard model consisting of the $t_{2g}$ manifold of $5d$ orbitals. To characterize the ground state, we introduce a local Kramer's doublet which can represent a state with effective angular momentum $J_{\rm eff}=|{\bm S}-{\bm L}|=1/2$ as well as spin $S=1/2$. Our systematic study of the pseudo-spin structure factor defined by the Kramer's doublet shows that the $J_{\rm eff}=1/2$ Mott insulator is smoothly connected to the $S=1/2$ Mott insulator. Using the Kramer's doublet as a variational state for the VCA, we examine the one-particle excitations for the Mott insulating phase. These results are compared with recent experiments on Sr$_2$IrO$_4$.

Variational cluster approximation study of Mott transition with strong spin-orbit coupling

Motivated by recent experiments on Sr2IrO4, the ground state magnetic and electronic structures are studied theoretically for a two-dimensional three-band Hubbard model with strong spin-orbit coupling. To treat spin-orbit coupling, local Coulomb interactions, and band structure effects on the same footing, the variational cluster approximation based on the self-energy functional theory is employed. It is found that for a relatively large coupling region, the ground state is an anisotropic antiferromagnetic Mott insulator of an effective local angular momentum Jeff = 1/2 with xy plane as an easy plane. This anisotropy is caused by the strong spin-orbit coupling along with the inter-orbital Hund's coupling. The momentum resolved one-particle excitations are also studied for the Mott insulating phase. It is found that the low-energy one-particle excitations consist mostly of the Jeff = 1/2 state, a direct evidence of a novel Jeff = 1/2 Mott insulator.

Microscopic Study of a Spin-Orbit-Induced Mott Insulator in Ir Oxides

Motivated by recent experiments of a novel 5d Mott insulator in Sr2IrO4, we have studied the two-dimensional three-orbital Hubbard model with a spin-orbit coupling λ. The variational Monte Carlo method is used to obtain the ground state phase diagram with varying an on-site Coulomb interaction U as well as λ. It is found that the transition from a paramagnetic metal to an antiferromagnetic insulator occurs at a finite U=U(MI), which is greatly reduced by a large λ, characteristic of 5d electrons, and leads to the "spin-orbit-induced" Mott insulator. It is also found that the Hund's coupling induces the anisotropic spin exchange and stabilizes the in-plane antiferromagnetic order. We have further studied the one-particle excitations by using the variational cluster approximation and revealed the internal electronic structure of this novel Mott insulator. These findings are in agreement with experimental observations on Sr2IrO4.

Importance of local correlations for the order parameter of high-Tcsuperconductors

It is shown that local temporal correlations in addition to non-local spatial correlations are important to understand the size and the doping dependence of the d-wave superconducting order parameter of high-temperature superconductors. To this end, the hole- and electron-doped two-dimensional Hubbard model at zero temperature is considered and treated by an extension of the variational cluster approximation. Within this approach, the effects of temporal correlations can be studied systematically and in a thermodynamically consistent way by comparing results obtained from different reference clusters. Contact can be made with previous cellular (plaquette) dynamical mean-field calculations. This shows that temporal correlations considerably decrease the order parameter and provide a substantial gain of binding energy. Besides, a few methodical insights regarding real-space quantum-cluster approaches are obtained in addition.

Spin-state transition inLaCoO3by variational cluster approximation

The variational cluster approximation is applied to the calculation of thermodynamical quantities and single-particle spectra of LaCoO3. Trial self-energies and the numerical value of the Luttinger-Ward functional are obtained by exact diagonalization of a CoO6 cluster. The VCA correctly predicts LaCoO3 as a paramagnetic insulator and a gradual and relatively smooth increase of the occupation of high-spin Co3+ ions causes the temperature dependence of entropy and magnetic susceptibility. The single particle spectral function agrees well with experiment, the experimentally observed temperature dependence of photoelectron spectra is reproduced satisfactorily. Remaining discrepancies with experiment highlight the importance of spin orbit coupling and local lattice relaxation.

Accessing thermodynamics from dynamical cluster-embedding approaches

Dynamical quantum-cluster approaches, such as different cluster extensions of the dynamical mean-field theory (cluster DMFT) or the variational cluster approximation (VCA), combined with efficient cluster solvers, such as the quantum Monte Carlo (QMC) method, provide controlled approximations of the single-particle Green's function for lattice models of strongly correlated electrons. To access the thermodynamics, however, a thermodynamical potential is needed. We present an efficient numerical algorithm to compute the grand potential within cluster-embedding approaches that are based on novel continuous-time QMC schemes. It is shown that the numerically exact cluster grand potential can be obtained from a quantum Wang-Landau technique to reweight the coefficients in the expansion of the partition function. The lattice contributions to the grand potential are computed by a proper infinite summation over Matsubara frequencies. A proof of principle is given by applying the VCA to antiferromagnetic (short-range) order in the two-dimensional Hubbard model at finite temperatures.

Magnetic and metallic state at intermediate HubbardUcoupling in multiorbital models for undoped iron pnictides

Multiorbital Hubbard model Hamiltonians for the undoped parent compounds of the Fe-pnictide superconductors are investigated here using mean-field techniques. For a realistic four-orbital model, our results show the existence of an intermediate Hubbard $U$ coupling regime where the mean-field ground state has a $(\ensuremath{\pi},0)$ antiferromagnetic order, as in neutron-scattering experiments, while remaining metallic due to the phenomenon of band overlaps. The angle-resolved photoemission intensity and Fermi surface of this magnetic and metallic state are discussed. Other models are also investigated, including a two-orbital model where not only the mean-field technique can be used but also the exact diagonalization in small clusters and the variational cluster approximation in the bulk. The combined results of the three techniques point toward the existence of an intermediate-coupling magnetic and metallic state in the two-orbital model, similar to the intermediate-coupling mean-field state of the four-orbital model. We conclude that the state discussed here is compatible with the experimentally known properties of the undoped Fe pnictides.

Spectral properties of orbital polarons in Mott insulators

We address the spectral properties of Mott insulators with orbital degrees of freedom, and investigate cases where the orbital symmetry leads to Ising-type superexchange in the orbital sector. The paradigm of a hole propagating by its coupling to quantum fluctuations, known from the spin t-J model, then no longer applies. We find instead that when one of the two orbital flavors is immobile, as in the Falicov-Kimball model, trapped orbital polarons coexist with free hole propagation emerging from the effective three-site hopping in the regime of large on-site Coulomb interaction U. The spectral functions are found analytically in this case within the retraceable path approximation in one and two dimensions. On the contrary, when both of the orbitals are active, as in the model for t2g electrons in two dimensions, we find propagating polarons with incoherent scattering dressing the moving hole and renormalizing the quasiparticle dispersion. Here, the spectral func- tions, calculated using the self-consistent Born approximation, are anisotropic and depend on the orbital flavor. Unbiased conclusions concerning the spectral properties are established by comparing the above results for the orbital t-J models with those obtained using the variational cluster approximation or exact diagonalization for the corresponding Hubbard models. Themore » present work makes predictions concerning the essential features of photoemission spectra of certain fluorides and vanadates.« less

Correlated band structure of NiO, CoO, and MnO by variational cluster approximation

The variational cluster approximation proposed by Potthoff is applied to the calculation of the single-particle spectral function of the transition-metal oxides MnO, CoO, and NiO. Trial self-energies and the numerical value of the Luttinger-Ward functional are obtained by exact diagonalization of a ${\text{TMO}}_{6}$ cluster. The single-particle parameters of this cluster serve as variational parameters to construct a stationary point of the grand potential of the lattice system. The stationary point is found by a crossover procedure, which allows to go continuously from an array of disconnected clusters to the lattice system. The self-energy is found to contain irrelevant degrees of freedom, which have marginal impact on the grand potential and need to be excluded to obtain meaningful results. The obtained spectral functions are in good agreement with experimental data.

Hubbard Model on the Triangular Lattice: Spiral Order and Spin Liquid

We investigate the half-filled Hubbard model on an isotropic triangular lattice with the variational cluster approximation. By decreasing the on-site repulsion U (or equivalently increasing pressure) we go from a phase with long-range, three-sublattice, spiral magnetic order, to a nonmagnetic Mott insulating phase--a spin liquid--and then, for U less or similar to 6.7t, to a metallic phase. Clusters of sizes 3, 6, and 15 with open boundary conditions are used in these calculations, and an extrapolation to infinite size is argued to lead to a disordered phase at U = 8t, but to a spiral order at U greater or similar to 12.

Magnetism andd-wave superconductivity on the half-filled square lattice with frustration

The role of frustration and interaction strength on the half-filled Hubbard model is studied on the square lattice with nearest- and next-nearest-neighbor hoppings $t$ and ${t}^{\ensuremath{'}}$ using the variational cluster approximation (VCA). At half-filling, we find two phases with long-range antiferromagnetic (AF) order: the usual N\'eel phase, stable at small frustration ${t}^{\ensuremath{'}}∕t$, and the so-called collinear (or superantiferromagnet) phase with ordering wave vector $(\ensuremath{\pi},0)$ or $(0,\ensuremath{\pi})$, stable for large frustration. These are separated by a phase with no detectable long-range magnetic order. We also find the $d$-wave superconducting (SC) phase $({d}_{{x}^{2}\ensuremath{-}{y}^{2}})$, which is favored by frustration if it is not too large. Intriguingly, there is a broad region of coexistence where both AF and SC order parameters have nonzero values. In addition, the physics of the metal-insulator transition in the normal state is analyzed. The results obtained with the help of the VCA method are compared with the large-$U$ expansion of the Hubbard model and known results for the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model. These results are relevant for pressure studies of undoped parents of the high-temperature superconductors: we predict that an insulator to $d$-wave SC transition may appear under pressure.

Convexity of the self-energy functional in the variational cluster approximation

In the variational cluster approximation (VCA) (or variational cluster perturbation theory), widely used to study the Hubbard model, a fundamental problem that renders variational solutions difficult in practice is its known lack of convexity at stationary points, i.e., the physical solutions can be saddle points rather than extrema of the self-energy functional. Here, we suggest two different approaches to construct a convex functional $\ensuremath{\Omega}[\ensuremath{\Sigma}]$. In the first approach, one can show analytically that in the approximation where the irreducible particle-hole vertex depends only on center of mass coordinates, the functional is convex away from phase transitions in the corresponding channel. Numerical tests on a tractable version of that functional show that convexity can be a nuisance when looking for instabilities both in the pairing and particle-hole channels. Therefore, an alternative phenomenological functional is proposed. Convexity is explicitly enforced only with respect to a restricted set of variables, such as the cluster chemical potential that is known to be otherwise problematic. Numerical tests show that our functional is convex at the physical solutions of VCA and allows second-order phase transitions in the pairing channel as well. This opens the way to the use of more efficient algorithms to find solutions of the VCA equations.

From cluster to solid: Variational cluster approximation applied toNiO

The variational cluster approximation is applied to the calculation of the single-particle spectral function of $\mathrm{NiO}$. Trial self-energies and the numerical value of the Luttinger-Ward functional are obtained by exact diagonalization of ${\mathrm{NiO}}_{6}$ clusters, and the single-particle parameters of the clusters serve as variational parameters to obtain a stationary point of the grand potential of the lattice system. Good agreement with experiment is obtained.

Pseudogap and high-temperature superconductivity from weak to strong coupling. Towards a quantitative theory (Review Article)

This is a short review of the theoretical work on the two-dimensional Hubbard model performed in Sherbrooke in the last few years. It is written on the occasion of the twentieth anniversary of the discovery of high-temperature superconductivity. We discuss several approaches, how they were benchmarked and how they agree sufficiently with each other that we can trust that the results are accurate solutions of the Hubbard model. Then comparisons are made with experiment. We show that the Hubbard model does exhibit d-wave superconductivity and antiferromagnetism essentially where they are observed for both hole- and electron-doped cuprates. We also show that the pseudogap phenomenon comes out of these calculations. In the case of electron-doped high temperature superconductors, comparisons with angle-resolved photoemission experiments are nearly quantitative. The value of the pseudogap temperature observed for these compounds in recent photoemission experiments had been predicted by theory before it was observed experimentally. Additional experimental confirmation would be useful. The theoretical methods that are surveyed include mostly the two-particle self-consistent approach, variational cluster perturbation theory (or variational cluster approximation), and cellular dynamical mean-field theory.

Extended variational cluster approximation for correlated systems

The variational cluster approximation (VCA) proposed by M. Potthoff et al. [Phys. Rev. Lett. 91, 206402 (2003)] is extended to electron or spin systems with nonlocal interactions. By introducing more than one source field in the action and employing the Legendre transformation, we derive a generalized self-energy functional with stationary properties. Applying this functional to a proper reference system, we construct the extended VCA (EVCA). In the limit of continuous degrees of freedom for the reference system, EVCA can recover the cluster extension of the extended dynamical mean-field theory (EDMFT). For a system with correlated hopping, the EVCA recovers the cluster extension of the dynamical mean-field theory for correlated hopping. Using a discrete reference system composed of decoupled three-site single impurities, we test the theory for the extended Hubbard model. Quantitatively good results as compared with EDMFT are obtained. We also propose VCA (EVCA) based on clusters with periodic boundary conditions. It has the (extended) dynamical cluster approximation as the continuous limit. A number of related issues are discussed.

Theory of Biaxial Nematic Ordering in Rod-Disc Mixtures Revisited

We use the variational cluster approximation to study the relative thermodynamic stability of the spatially uniform phases of binary mixtures of hard rods and discs. The factors promoting the stability of the biaxial nematic phase are identified and discussed. The results suggest that a stable thermotropic nematic biaxial mixture cannot be obtained from molecules of the sizes and electric dipole interaction strengths commonly encountered in real calamitic and discotic thermotropic phases.