The realization of antiferromagnetic (AF) correlations in ultracold fermionic atoms on an optical lattice is a significant achievement. Experiments have been carried out in one, two, and three dimensions, and have also studied anisotropic configurations with stronger tunneling in some lattice directions. Such anisotropy is relevant to the physics of cuprate superconductors and other strongly correlated materials. Moreover, this anisotropy might be harnessed to enhance AF order. Here we numerically investigate, using the determinant quantum Monte Carlo method, a simple realization of anisotropy in the three-dimensional (3D) Hubbard model in which the tunneling between planes, ${t}_{\ensuremath{\perp}}$, is unequal to the intraplane tunneling $t$. This model interpolates between the three-dimensional isotropic (${t}_{\ensuremath{\perp}}=t$) and two-dimensional (2D; ${t}_{\ensuremath{\perp}}=0$) systems. We show that at fixed interaction strength to tunneling ratio ($U/t$), anisotropy can enhance the magnetic structure factor relative to both 2D and 3D results. However, this enhancement occurs at interaction strengths below those for which the N\'eel temperature ${T}_{\mathrm{N}\stackrel{\ifmmode \acute{}\else \'{}\fi{}}{\mathrm{e}}\mathrm{el}}$ is largest, in such a way that the structure factor cannot be made to exceed its value in isotropic 3D systems at the optimal $U/t$. We characterize the 2D-3D crossover in terms of the magnetic structure factor, real space spin correlations, number of doubly occupied sites, and thermodynamic observables. An interesting implication of our results stems from the entropy's dependence on anisotropy. As the system evolves from 3D to 2D, the entropy at a fixed temperature increases. Correspondingly, at fixed entropy, the temperature will decrease going from 3D to 2D. This suggests a cooling protocol in which the dimensionality is adiabatically changed from 3D to 2D.

Thermodynamics and magnetism in the two-dimensional to three-dimensional crossover of the Hubbard model

Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.

http://absimage.aps.org/image/MAR16/MWS_MAR16-2015-020101.pdf

Solutions of the Two Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms

With the purpose of investigating coexistence between magnetic order and superconductivity, we consider a model in which conduction electrons interact with each other, via an attractive Hubbard on-site coupling U, and with local moments on every site, via a Kondo-like coupling, J. The model is solved on a simple cubic lattice through a Hartree-Fock approximation, within a 'semi-classical' framework which allows spiral magnetic modes to be stabilized. For a fixed electronic density, n c , the small J region of the ground state (T  =  0) phase diagram displays spiral antiferromagnetic (SAFM) states for small U. Upon increasing U, a state with coexistence between superconductivity (SC) and SAFM sets in; further increase in U turns the spiral mode into a Neel antiferromagnet. The large J region is a (singlet) Kondo phase. At finite temperatures, and in the region of coexistence, thermal fluctuations suppress the different ordered phases in succession: the SAFM phase at lower temperatures and SC at higher temperatures; also, reentrant behaviour is found to be induced by temperature. Our results provide a qualitative description of the competition between local moment magnetism and superconductivity in the borocarbides family.

A mean-field approach to Kondo-attractive-Hubbard model.

We present a study of the influence of disorder on the Mott metal-insulator transition for the organic charge-transfer salt $\kappa$-(BEDT-TTF)$_2$Cu[N(CN)$_2$]Cl. To this end, disorder was introduced into the system in a controlled way by exposing the single crystals to x-ray irradiation. The crystals were then fine-tuned across the Mott transition by the application of continuously controllable He-gas pressure at low temperatures. Measurements of the thermal expansion and resistance show that the first-order character of the Mott transition prevails for low irradiation doses achieved by irradiation times up to 100 h. For these crystals with a moderate degree of disorder, we find a first-order transition line which ends in a second-order critical endpoint, akin to the pristine crystals. Compared to the latter, however, we observe a significant reduction of both, the critical pressure $p_c$ and the critical temperature $T_c$. This result is consistent with the theoretically-predicted formation of a soft Coulomb gap in the presence of strong correlations and small disorder. Furthermore, we demonstrate, similar to the observation for the pristine sample, that the Mott transition after 50 h of irradiation is accompanied by sizable lattice effects, the critical behavior of which can be well described by mean-field theory. Our results demonstrate that the character of the Mott transition remains essentially unchanged at a low disorder level. However, after an irradiation time of 150 h, no clear signatures of a discontinuous metal-insulator transition could be revealed anymore. These results suggest that, above a certain disorder level, the metal-insulator transition becomes a smeared first-order transition with some residual hysteresis.

Effects of Disorder on the Pressure-Induced Mott Transition in $\kappa$-BEDT-TTF)$_2$Cu[N(CN)$_2$]Cl

We present a study of the influence of disorder on the Mott metal-insulator transition for the organic charge-transfer salt κ -(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl. To this end, disorder was introduced into the system in a controlled way by exposing the single crystals to X-ray irradiation. The crystals were then fine-tuned across the Mott transition by the application of continuously controllable He-gas pressure at low temperatures. Measurements of the thermal expansion and resistance show that the first-order character of the Mott transition prevails for low irradiation doses achieved by irradiation times up to 100 h. For these crystals with a moderate degree of disorder, we find a first-order transition line which ends in a second-order critical endpoint, akin to the pristine crystals. Compared to the latter, however, we observe a significant reduction of both, the critical pressure p c and the critical temperature T c . This result is consistent with the theoretically-predicted formation of a soft Coulomb gap in the presence of strong correlations and small disorder. Furthermore, we demonstrate, similar to the observation for the pristine sample, that the Mott transition after 50 h of irradiation is accompanied by sizable lattice effects, the critical behavior of which can be well described by mean-field theory. Our results demonstrate that the character of the Mott transition remains essentially unchanged at a low disorder level. However, after an irradiation time of 150 h, no clear signatures of a discontinuous metal-insulator transition could be revealed anymore. These results suggest that, above a certain disorder level, the metal-insulator transition becomes a smeared first-order transition with some residual hysteresis.

https://www.mdpi.com/2073-4352/8/1/38/pdf/1

Effects of Disorder on the Pressure-Induced Mott Transition in κ-(BEDT-TTF)2Cu[N(CN)2]Cl

We present evidence for Mott quantum criticality in an anisotropic two-dimensional system of coupled Hubbard chains at half-filling. In this scenario emerging from variational cluster approximation and cluster dynamical mean-field theory, the interchain hopping t_{⊥} acts as a control parameter driving the second-order critical end point T_{c} of the metal-insulator transition down to zero at t_{⊥}^{c}/t≃0.2. Below t_{⊥}^{c}, the volume of the hole and electron Fermi pockets of a compensated metal vanishes continuously at the Mott transition. Above t_{⊥}^{c}, the volume reduction of the pockets is cut off by a first-order transition. We discuss the relevance of our findings to a putative quantum critical point in layered organic conductors, whose location remains elusive so far.

Mott Quantum Criticality in the Anisotropic 2D Hubbard Model.

We investigate in detail antiferromagnetic (AF) and superconducting (SC) phases as well as their coexistence in the two-dimensional Kondo lattice model on a square lattice, which is a paradigmatic model for heavy-fermion materials. The results presented are mainly obtained using the variational cluster approximation (VCA) and are complemented by analytical findings for the equations of motion of pairing susceptibilities. A particularly interesting aspect is the possibility to have $s$-wave SC near half-filling as reported by Bodensiek et al. [Phys. Rev. Lett. 110, 146406 (2013)] When doping the system, we identify three regions which correspond to an AF metallic phase with small Fermi surface at weak coupling, an AF metal with a different Fermi surface topology at intermediate coupling, and a paramagnetic metal with a large Fermi surface at strong coupling. The transition between these two AF phases is found to be discontinuous at lower fillings, but turns to a continuous one when approaching half-filling. In the quest for $s$-wave superconductivity, only solutions are found which possess mean-field character. No true superconducting solutions caused by correlation effects are found in the $s$-wave channel. In contrast, we clearly identify robust $d$-wave pairing away from half-filling. However, we show that only by treating antiferromagnetism and superconductivity on equal footing, artificial superconducting solutions at half-filling can be avoided. Our VCA findings support scenarios previously identified by variational Monte Carlo approaches and are a starting point for future investigations with VCA and further approaches such as cluster-embedding methods.

Variational cluster approach to superconductivity and magnetism in the Kondo lattice model

Motivated by dimensional crossover in layered organic $\ensuremath{\kappa}$ salts, we determine the phase diagram of a system of four periodically coupled Hubbard chains with frustration at half filling as a function of the interchain hopping ${t}_{\ensuremath{\perp}}/t$ and interaction strength $U/t$ at a fixed ratio of frustration and interchain hopping ${t}^{\ensuremath{'}}/{t}_{\ensuremath{\perp}}=\ensuremath{-}0.5$. We cover the range from the one-dimensional limit of uncoupled chains (${t}_{\ensuremath{\perp}}/t=0.0$) to the isotropic model (${t}_{\ensuremath{\perp}}/t=1.0$). For strong $U/t$, we find an antiferromagnetic insulator; in the weak-to-moderate-interaction regime, the phase diagram features quasi-one-dimensional antiferromagnetic behavior, an incommensurate spin density wave, and a metallic phase as ${t}_{\ensuremath{\perp}}/t$ is increased. We characterize the phases through their magnetic ordering, dielectric response, and dominant static correlations. Our analysis is based primarily on a variant of the density-matrix renormalization-group algorithm based on an efficient hybrid--real-momentum-space formulation, in which we can treat relatively large lattices albeit of a limited width. This is complemented by a variational cluster approximation study with a cluster geometry corresponding to the cylindrical lattice allowing us to directly compare the two methods for this geometry. As an outlook, we make contact with work studying dimensional crossover in the full two-dimensional system.

Anisotropy crossover in the frustrated Hubbard model on four-chain cylinders

Variational Cluster Approach to Superconductivity and Magnetism in the Kondo Lattice Model

/pdf/mott-quantum-criticality-in-the-anisotropic-2d-hubbard-model-5b65azfynx.pdf

Benjamin Lenz

Papers

Mott Quantum Criticality in the Anisotropic 2D Hubbard Model.

Variational cluster approach to superconductivity and magnetism in the Kondo lattice model

Anisotropy crossover in the frustrated Hubbard model on four-chain cylinders

Variational Cluster Approach to Superconductivity and Magnetism in the Kondo Lattice Model

Mott Quantum Criticality in the Anisotropic 2D Hubbard Model