One-band Hubbard Model Research Articles

Mott transition and magnetism on the anisotropic triangular lattice

Spin-liquid behavior was recently suggested experimentally in the moderately one-dimensional organic compound $\ensuremath{\kappa}\text{\ensuremath{-}}{\mathrm{H}}_{3}{(\text{Cat-EDT-TTF})}_{2}$. This compound can be modeled by the one-band Hubbard model on the anisotropic triangular lattice with ${t}^{\ensuremath{'}}/t\ensuremath{\simeq}1.5$, where ${t}^{\ensuremath{'}}$ is the minority hopping. It thus becomes important to extend previous studies, that were performed in the range $0\ensuremath{\le}{t}^{\ensuremath{'}}/t\ensuremath{\le}1.2$, to find out whether there is a regime where Mott insulating behavior can be found without long-range magnetic order. To this end, we study the above model in the range $1.2\ensuremath{\le}{t}^{\ensuremath{'}}/t\ensuremath{\le}2$ using cluster dynamical mean-field theory (CDMFT). We argue that it is important to choose a symmetry-preserving cluster rather than a quasi-one-dimensional cluster. We find that, upon increasing ${t}^{\ensuremath{'}}/t$ beyond ${t}^{\ensuremath{'}}/t\ensuremath{\approx}1.3$, the Mott transition at zero temperature is replaced by a first-order transition separating a metallic state from a collinear magnetic insulating state excluding the possibility to find a quantum spin liquid for the physically relevant value ${t}^{\ensuremath{'}}/t\ensuremath{\simeq}1.5$. The phase diagram obtained in this study can provide a working basis for moderately one-dimensional compounds on the anisotropic triangular lattice.

Hidden Mott transition and large- U superconductivity in the two-dimensional Hubbard model

We consider the one-band Hubbard model on the square lattice by using variational and Green's function Monte Carlo methods, where the variational states contain Jastrow and backflow correlations on top of an uncorrelated wave function that includes BCS pairing and magnetic order. At half filling, where the ground state is antiferromagnetically ordered for any value of the on-site interaction $U$, we can identify a hidden critical point $U_{\rm Mott}$, above which a finite BCS pairing is stabilized in the wave function. The existence of this point is reminiscent of the Mott transition in the paramagnetic sector and determines a separation between a Slater insulator (at small values of $U$), where magnetism induces a potential energy gain, and a Mott insulator (at large values of $U$), where magnetic correlations drive a kinetic energy gain. Most importantly, the existence of $U_{\rm Mott}$ has crucial consequences when doping the system: We observe a tendency to phase separation into a hole-rich and a hole-poor region only when doping the Slater insulator, while the system is uniform by doping the Mott insulator. Superconducting correlations are clearly observed above $U_{\rm Mott}$, leading to the characteristic dome structure in doping. Furthermore, we show that the energy gain due to the presence of a finite BCS pairing above $U_{\rm Mott}$ shifts from the potential to the kinetic sector by increasing the value of the Coulomb repulsion.

Antagonistic effects of nearest-neighbor repulsion on the superconducting pairing dynamics in the doped Mott insulator regime

The nearest-neighbor superexchange-mediated mechanism for d_{x^2-y^2}-wave superconductivity in the one-band Hubbard model faces the challenge that nearest-neighbor Coulomb repulsion can be larger than superexchange. To answer this question, we use cellular dynamical mean-field theory (CDMFT) with a continuous-time quantum Monte Carlo solver to determine the superconducting phase diagram as a function of temperature and doping for on-site repulsion $U=9t$ and nearest-neighbor repulsion $V=0,2t,4t$. In the underdoped regime, $V$ increases the CDMFT superconducting transition temperature $T_c^d$ even though it decreases the superconducting order parameter at low temperature for all dopings. However, $V$ decreases $T_c^d$ in the overdoped regime. We gain insight into these paradoxical results through a detailed study of the frequency dependence of the anomalous spectral function, extracted at finite temperature via the MaxEntAux method for analytic continuation. A systematic study of dynamical positive and negative contributions to pairing reveals that even though $V$ has a high-frequency depairing contribution, it also has a low frequency pairing contribution since it can reinforce superexchange through $J=4t^2/(U-V)$. Retardation is thus crucial to understand pairing in doped Mott insulators, as suggested by previous zero-temperature studies. We also comment on the tendency to charge order for large $V$ and on the persistence of d-wave superconductivity over extended-$s$ or s+d-wave.

Mott transitions in the periodic Anderson model

The periodic Anderson model (PAM) is studied within the framework of dynamical mean-field theory, with particular emphasis on the interaction-driven Mott transition it contains, and on resultant Mott insulators of both Mott–Hubbard and charge-transfer type. The form of the PAM phase diagram is first deduced on general grounds using two exact results, over the full range of model parameters and including metallic, Mott, Kondo and band insulator phases. The effective low-energy model which describes the PAM in the vicinity of a Mott transition is then shown to be a one-band Hubbard model, with effective hoppings that are not in general solely nearest neighbour, but decay exponentially with distance. This mapping is shown to have a range of implications for the physics of the problem, from phase boundaries to single-particle dynamics; all of which are confirmed and supplemented by NRG calculations. Finally we consider the locally degenerate, non-Fermi liquid Mott insulator, to describe which requires a two-self-energy description. This is shown to yield a number of exact results for the associated local moment, charge, and interaction-renormalised levels, together with a generalisation of Luttinger’s theorem to the Mott insulator.

Towards TDDFT for Strongly Correlated Materials

We present some details of our recently-proposed Time-Dependent Density-Functional Theory (TDDFT) for strongly-correlated materials in which the exchange-correlation (XC) kernel is derived from the charge susceptibility obtained using Dynamical Mean-Field Theory (the TDDFT + DMFT approach). We proceed with deriving the expression for the XC kernel for the one-band Hubbard model by solving DMFT equations via two approaches, the Hirsch–Fye Quantum Monte Carlo (HF-QMC) and an approximate low-cost perturbation theory approach, and demonstrate that the latter gives results that are comparable to the exact HF-QMC solution. Furthermore, through a variety of applications, we propose a simple analytical formula for the XC kernel. Additionally, we use the exact and approximate kernels to examine the nonhomogeneous ultrafast response of two systems: a one-band Hubbard model and a Mott insulator YTiO3. We show that the frequency dependence of the kernel, i.e., memory effects, is important for dynamics at the femtosecond timescale. We also conclude that strong correlations lead to the presence of beats in the time-dependent electric conductivity in YTiO3, a feature that could be tested experimentally and that could help validate the few approximations used in our formulation. We conclude by proposing an algorithm for the generalization of the theory to non-linear response.

Superconducting Phase Diagram of the Paramagnetic One-Band Hubbard Model

We study spin-fluctuation-mediated superconductivity in the one-band Hubbard model. Higher-order effective interactions in U give rise to a superconducting instability which is very sensitive to changes in the Fermi surface topology arising as a function of doping and changes in the band structure. We show the superconducting phase diagram as a function of doping and next-nearest neighbor hopping in the limit of very small Coulomb interaction strength and discuss peculiarities arising at the phase boundaries separating different superconducting domains.

Acoustic magnons in the long-wavelength limit: Investigating the Goldstone violation in many-body perturbation theory

Collective spin excitations in magnetic materials arise from the correlated motion of electron-hole pairs with opposite spins. The pair propagation is described by the transverse magnetic susceptibility, which we calculate within many-body perturbation theory from first principles employing the full-potential linearized augmented-plane-wave formalism. Ferromagnetic materials exhibit a spontaneously broken global rotation symmetry in spin space leading to the appearance of acoustic magnons (zero gap) in the long-wavelength limit. However, due to approximations used in the numerical scheme, the acoustic magnon dispersion exhibits a small but finite gap at $\mathrm{\ensuremath{\Gamma}}$. We analyze this violation of the Goldstone mode and present an approach that implements the magnetic susceptibility using a renormalized Green function instead of the Kohn-Sham one. This much more expensive approach shows substantial improvement of the Goldstone-mode condition. In addition, we discuss a possible correction scheme, which involves an adjustment of the Kohn-Sham exchange splitting, which is motivated by the spin-wave solution of the one-band Hubbard model. The new exchange splittings turn out to be closer to experiment. We present corrected magnon spectra for the elementary ferromagnets Fe, Co, and Ni.

Publisher's Note: Pairing symmetry of the one-band Hubbard model in the paramagnetic weak-coupling limit: A numerical RPA study [Phys. Rev. B92, 104505 (2015)

Publisher's Note: Pairing symmetry of the one-band Hubbard model in the paramagnetic weak-coupling limit: A numerical RPA study [Phys. Rev. B<b>92</b>, 104505 (2015)

Real-space Green&amp;rsquo;s function method for Fe-based superconductors with Cu substitution

We develop a novel real-space Green’s function method for the study of the Cu-substituted Fe-based superconductors based on the Green’s function equation of motion approach. As an extension of the Hubbard-I approximation for the one-band Hubbard model, a series of decoupling scheme are introduced for the multi-orbital Hubbard model. Since the real-space fluctuation induced by the disorder effect can be treated exactly, our method is very suitable for the study of the strongly correlated multi-orbital systems with the presence of impurities. An inhomogeneous three-orbital Hubbard model is employed to involve the electrons in the three dxy , dxz , and dxz orbitals of the iron and cupper atoms in Cu-substituted Fe-based superconductors. When the orbital-selective Mott transition happens in the iron-based superconductors, a Mott gap can be found in the dxy orbital, while the degenerated d xz / yz orbitals remain metallic. With the increase of the Cu substitution, the dopants will induce in-gap states near the Fermi level within the Mott gap of the dxy orbital, and the Mott gap can be fulfilled by the in-gap states at higher doping cases. In addition, we find that the in-gap states only appear in a few discrete lattice points, where the occupancy of dxy orbital is away from half filling. It is also found that all these electronic states are Anderson localized. Meanwhile, the orbital polarization of the whole system increases with the enhancement of the real space fluctuation of the orbital occupancy induced by the Cu substitution. We also find that the Hund’s rule coupling has strong effect in suppressing the real space fluctuation, which causes the in-gap states to vanish. As a result, the dxy orbital will restore to Mott-Hubbard insulator, suggesting that the critical value J c of Hund’s rule coupling for the Mott-Hubbard transition will be pushed up significantly.

Fermionic spinon and holon statistics in the pyrochlore quantum spin liquid

The one-band Hubbard model on the pyrochlore lattice contains an extended quantum spin-liquid phase formed from the manifold of singlet dimer coverings. We demonstrate that the massive and deconfined spinon excitations of this system have fermionic statistics. Holonic quasiparticles introduced by doping are also fermions and we explain the origin of this counterintuitive result.

Mott insulators and the doping-induced Mott transition within DMFT: exact results for the one-band Hubbard model

The paramagnetic phase of the one-band Hubbard model is studied at zero-temperature, within the framework of dynamical mean-field theory, and for general particle-hole asymmetry where a doping-induced Mott transition occurs. Our primary focus is the Mott insulator (MI) phase, and our main aim to establish what can be shown exactly about it. To handle the locally doubly-degenerate MI requires two distinct self-energies, which reflect the broken symmetry nature of the phase and together determine the standard single self-energy. Exact results are obtained for the local charge, local magnetic moment and associated spin susceptibilities, the interaction-renormalised levels, and the low-energy behaviour of the self-energy in the MI phase. The metallic phase is also considered briefly, and shown to acquire an emergent particle-hole symmetry as the Mott transition is approached. Throughout the metal, Luttinger’s theorem is reflected in the vanishing of the Luttinger integral; for the generic MI by contrast this is shown to be non-vanishing, but again to have a universal magnitude. Numerical results are also obtained using NRG, for the metal/MI phase boundary, the scaling behaviour of the charge as the Mott transition is aproached from the metal, and associated universal scaling of single-particle dynamics as the low-energy Kondo scale vanishes.

Coexistence and competition of on-site and intersite Coulomb interactions in Mott-molecular-dimers

Abstract We reveal the interplay between on-site ( U ) and intersite ( V ) Coulomb interactions in the extended two-site Hubbard model. Due to its atomic-like form quantum correlations intrinsic to Mott-molecular-dimers are exactly computed. Our results for physical quantities such as double occupancy and specific heat are consistent with those obtained for the one-band Hubbard model, suggesting that a two-site dimer model is able to capture the essential thermodynamic properties of strongly interacting electron systems. It is noted that intersite Coulomb interactions promote the formation of doublons, which compete with the spin-singlet state induced by the on-site Coulomb repulsion. Our results are expected to be relevant for understanding electronic and thermodynamical properties of interacting electrons in systems with strongly coupled magnetic atoms.

A standard basis operator equation of motion impurity solver for dynamical mean field theory

We present an efficient impurity solver for the dynamical mean-field theory (DMFT). It is based on the separation of bath degrees of freedom into the low energy and the high energy parts. The former is solved exactly using exact diagonalization and the latter is treated approximately using Green's function equation of motion decoupling approximation. The two parts are combined coherently under the standard basis operator formalism. The impurity solver is applied to the Anderson impurity model and, combined with DMFT, to the one-band Hubbard model. Qualitative agreement is found with other well established methods. Some promising features and possible improvements of the present solver are discussed.

Pairing symmetry of the one-band Hubbard model in the paramagnetic weak-coupling limit: A numerical RPA study

We study the spin-fluctuation-mediated superconducting pairing gap in a weak-coupling approach to the Hubbard model for a two dimensional square lattice in the paramagnetic state. Performing a comprehensive theoretical study of the phase diagram as a function of filling, we find that the superconducting gap exhibits transitions from p-wave at very low electron fillings to d_{x^2-y^2}-wave symmetry close to half filling in agreement with previous reports. At intermediate filling levels, different gap symmetries appear as a consequence of the changes in the Fermi surface topology and the associated structure of the spin susceptibility. In particular, the vicinity of a van Hove singularity in the electronic structure close to the Fermi level has important consequences for the gap structure in favoring the otherwise sub-dominant triplet solution over the singlet d-wave solution. By solving the full gap equation, we find that the energetically favorable triplet solutions are chiral and break time reversal symmetry. Finally, we also calculate the detailed angular gap structure of the quasi-particle spectrum, and show how spin-fluctuation-mediated pairing leads to significant deviations from the first harmonics both in the singlet d_{x^2-y^2} gap as well as the chiral triplet gap solution.

D-wave superconductivity in coupled ladders

We study the one-band Hubbard model on the trellis lattice, a two-dimensional frustrated lattice of coupled two-leg ladders, with hopping amplitude $t$ within ladders and ${t}^{\ensuremath{'}}$ between ladders. For large $U/t$ this is a model for the cuprate ${\mathrm{Sr}}_{14\ensuremath{-}x}{\mathrm{Ca}}_{x}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$. We investigate the phase diagram as a function of doping for $U=10t$ using two quantum cluster methods: The variational cluster approximation (VCA), with clusters of sizes 8 and 12, and cellular dynamical mean field theory (CDMFT), both at zero temperature. Both methods predict a superconducting dome, ending at roughly 20% doping in VCA and 15% in CDMFT. In VCA, the superconducting order parameter is complex in a range of doping centered around 10%, corresponding to bulk chiral, $T$-violating superconductivity. However, the CDMFT solution is not chiral. We find evidence for a migration of the Cooper pairs from the interladder region towards the plaquettes as doping is increased.

Organization of the Hilbert space for exact diagonalization of Hubbard model

We present an alternative scheme to the widely used method of representing the basis of one-band Hubbard model through the relation I=I↑+2MI↓ given by Lin and Gubernatis (1993), where I↑, I↓ and I are the integer equivalents of binary representations of occupation patterns of spin up, spin down and both spin up and spin down electrons respectively, with M being the number of sites. We compute and store only I↑ or I↓ at a time to generate the full Hamiltonian matrix. The non-diagonal part of the Hamiltonian matrix given as I↓⊗H↑⊕H↓⊗I↑ is generated using a bottom-up approach by computing the small matrices H↑ (spin up hopping Hamiltonian) and H↓ (spin down hopping Hamiltonian) and then forming the tensor product with respective identity matrices I↓ and I↑, thereby saving significant computation time and memory. We find that the total CPU time to generate the non-diagonal part of the Hamiltonian matrix using the new one spin configuration basis scheme is reduced by about an order of magnitude as compared to the two spin configuration basis scheme. The present scheme is shown to be inherently parallelizable. Its application to translationally invariant systems, computation of Green’s functions and in impurity solver part of DMFT procedure is discussed and its extension to other models is also pointed out.

Testing the Monte Carlo–mean field approximation in the one-band Hubbard model

The canonical one-band Hubbard model is studied using a computational method that mixes the Monte Carlo procedure with the mean field approximation. This technique allows us to incorporate thermal fluctuations and the development of short-range magnetic order above ordering temperatures, contrary to the crude finite-temperature Hartree-Fock approximation, which incorrectly predicts a N\'eel temperature $T_N$ that grows linearly with the Hubbard $U/t$. The effective model studied here contains quantum and classical degrees of freedom. It thus belongs to the "spin fermion" model family widely employed in other contexts. Using exact diagonalization, supplemented by the traveling cluster approximation, for the fermionic sector, and classical Monte Carlo for the classical fields, the Hubbard $U/t$ vs. temperature $T/t$ phase diagram is studied employing large three and two dimensional clusters. We demonstrate that the method is capable of capturing the formation of local moments in the normal state without long-range order, the non-monotonicity of $T_N$ with increasing $U/t$, the development of gaps and pseudogaps in the density of states, and the two-peak structure in the specific heat. Extensive comparisons with determinant quantum Monte Carlo results suggest that the present approach is qualitatively, and often quantitatively, accurate, particularly at intermediate and high temperatures. Finally, we study the Hubbard model including plaquette diagonal hopping (i.e. the $t-t^\prime$ Hubbard model) in two dimensions and show that our approach allows us to study low temperature properties where determinant quantum Monte Carlo fails due to the fermion sign problem. Future applications of this method include multi-orbital Hubbard models such as those needed for iron-based superconductors.

A VARIATIONAL THEORY OF QUASI-PARTICLES IN A 3D N x N x N CUBIC LATTICE

The single-band Hubbard Hamiltonian study faces a serious limitation and difficulty as we move away from finite - size lattices to larger N - dimensional lattices. Thus there is the needto develop the means of overcoming the finite - size lattice defects as we pass on to a higher dimension.In this work, a quantitative approximation to the one-band Hubbard model is presented using a variational analytic approach. The goal of this work, therefore, is to explore quantitatively the lowest ground-state energy and the pairing correlations in 3D N x N x N lattices of the Hubbard model. We developed the unit step model as an approximate solution to the single-band Hubbard Hamiltonian to solve variationallythe correlation of two interacting elections on a three-dimensional cubic lattice. We also showed primarily how to derive possible electronic states available for several even and odd3D lattices, although, this work places more emphasis on a 3D 5 x 5 x 5 lattice. The results emerging from our present study compared favourablywith the results of Gutzwillervariational approach (GVA) and correlated variational approach (CVA), at thelarge limit of the Coulomb interaction strength (U/4t). It is revealed in this study, that the repulsive Coulomb interaction which in part leads to the strong electronic correlations, would indicate that the two electron system prefer not to condense into s-wave superconducting singlet state (s = 0), at high positive values of the interaction strength.

Hubbard Model on the Pyrochlore Lattice: A 3D Quantum Spin Liquid

We demonstrate that the insulating one-band Hubbard model on the pyrochlore lattice contains, for realistic parameters, an extended quantum spin-liquid phase. This is a three-dimensional spin liquid formed from a highly degenerate manifold of dimer-based states, which is a subset of the classical dimer coverings obeying the ice rules. It possesses spinon excitations, which are both massive and deconfined, and on doping it exhibits spin-charge separation. We discuss the realization of this state in effective S = 1/2 pyrochlore materials with and without spin-orbit coupling.

Ergodicity of the hybridization-expansion Monte Carlo algorithm for broken-symmetry states

With the success of dynamical mean-field theories, solvers for quantum-impurity problems have become an important tool for the numerical study of strongly correlated systems. Continuous-time quantum Monte Carlo sampling of the expansion in powers of the hybridization between the ``impurity'' and the bath provides a powerful solver when interactions are strong. Here we show that the usual updates that add or remove a pair of creation-annihilation operators are rigorously not ergodic for several classes of broken symmetries that involve spatial components. We show that updates with larger numbers of simultaneous updates of pairs of creation-annihilation operators remedy this problem. As an example, we apply the four-operator updates that are necessary for ergodicity to the case of $d$-wave superconductivity in plaquette dynamical mean-field theory for the one-band Hubbard model. While the results are qualitatively similar to those previously published, they are quantitatively better than previous ones, being closer to those obtained by other approaches.