The charge degrees of freedom in several different organic charge transfer salts display slow or glassy dynamics. In order to gain insight into this behaviour, we obtain the low energy theory for an extended dimer Hubbard model, taking into account the occupations of sites on neighbouring dimers. We take a classical limit of the resulting effective model of coupled spins and dimers and study it using classical Monte Carlo simulations. We find that frustration induced by intra- and inter-dimer interactions leads to glassiness in the charge degress of freedom in the absence of ordering of the spin degrees of freedom. Our results may have relevance to experimental observations of relaxor ferroelectric behaviour in the dynamics of organic charge transfer salts.
We numerically investigate the phase diagram of two-dimensional site-diluted coupled dimer systems in an external magnetic field. We show that this phase diagram is characterized by the presence of an extended Bose glass, not accessible to mean-field approximation, and stemming from the localization of two distinct species of bosonic quasiparticles appearing in the ground state. On the one hand, non-magnetic impurities doped into the dimer-singlet phase of a weakly coupled dimer system are known to free up local magnetic moments. The deviations of these local moments from full polarization alo…
We derive an effective Hamiltonian for the ionic Hubbard model at half filling, extended to include nearest-neighbor repulsion. Using a spin-particle transformation, the effective model is mapped onto simple spin-1 models in two particular cases. Using another spin-particle transformation, a slightly modified model is mapped into an SU(3) antiferromagnetic Heisenberg model whose exact ground state is known to be spontaneously dimerized. From the effective models several properties of the dimerized phase are discussed, like ferroelectricity and fractional charge excitations. Using bosonization …
We consider a formulation of the equation of motion technique for Green's function in which the unknown averages are computed by solving a linear system. This linear system appears solvable for all finite temperatures, but depending on the system parameters the condition number can be very large, making the solution numerically unfeasible at low temperatures. In the example that we consider, the Hubbard dimer, we can get rid of this problem by making use of total spin as a good quantum number.
We use the Kadanoff-Baym equations within the framework of many-body perturbation theory to study the double occupancies in a Hubbard dimer. The double occupancies are obtained from the equations of motion of the single-particle Green's function. Our calculations show that the approximate double occupancies can become negative. This is a shortcoming of approximate, and yet conserving, many-body self-energies. Among the tested perturbation schemes, the T-matrix approximation is the only one providing double occupancies which are always positive.
We study the one-dimensional extended Hubbard model with alternating size of the hopping integrals using the density-matrix renormalization group method. We calculate the spin gap, the Tomonaga-Luttinger parameter, and the charge-density-wave order parameter for various dimerizations, interaction strengths, and band fillings. At half band-filling the spin and charge excitations are gapped but these gaps disappear for infinitesimal hole doping. At quarter filling, the Umklapp scattering in the half-filled lower Peierls band generates a gap for the charge excitations but the gapless spin excitat…
The spectral potential is the dynamical generalization of the Kohn-Sham potential. It targets, in principle exactly, the spectral function in addition to the electronic density. Here we examine the spectral potential in one of the simplest solvable models exhibiting a non-trivial interplay between electron-electron interaction and inhomogeneity, namely the asymmetric Hubbard dimer. We discuss a general strategy to introduce approximations, which consists in calculating the spectral potential in the homogeneous limit (here represented by the symmetric Hubbard dimer) and importing it in the real…
We use a density matrix renormalization group method to study quantitatively the phase diagram of a one-dimensional extended Hubbard model at half-filling by investigating the correlation functions and structure factors. We confirm the existence of a novel narrow region with long-rang bond-order-wave order which is highly controversial recently between the charge-density-wave phase and Mott insulator phase. We determined accurately the position of the tricritical point $U_t\simeq 7.2t$, $V_t\simeq 3.746t$ which is quite different from previous studies.
We formulate the Hubbard model for the simple cubic lattice in the representation of interacting dimers applying the exact solution of the dimer problem. By eliminating from the considerations unoccupied dimer energy levels in the large U limit (it is the only assumption) we analytically derive the Hubbard Hamiltonian for the dimer (analogous to the well-known t-J model), as well as, the Hubbard Hamiltonian for the crystal as a whole by means of the projection technique. Using this approach we can better visualize the complexity of the model, so deeply hidden in its original form. The resultin…
The Dynamic Hubbard Model represents the physics of a multi-band Hubbard model by using a pseudo-spin degree of freedom to dynamically modify the on-site Coulomb interaction. Here we use a dimer system to obtain analytical results for this model. The spectral function and the optical conductivity are calculated analytically for any number of electrons, and the distribution of optical spectral weight is analyzed in great detail. The impact of polaron-like effects due to overlaps between pseudo-spin states on the optical spectral weight distribution is derived analytically. Our conclusions suppo…
In this paper, we introduce and study the fermionic concurrence in a two-site extended Hubbard model. Its behaviors both at the ground state and finite temperatures as function of Coulomb interaction $U$ (on-site) and $V$ (nearest-neighbor) are obtained analytically and numerically. We also investigate the change of the concurrence under a nonuniform field, including local potential and magnetic field, and find that the concurrence can be modulated by these fields.
We examine the time evolution of an asymmetric Hubbard dimer, which has a different on-site interaction on the two sites. The Hamiltonian has a time-dependent hopping term, which can be employed to describe an electric field (which creates a Hamiltonian with complex matrix elements), or it can describe a modulation of the lattice (which has real matrix elements). By examining the symmetries under spin and pseudospin, we show that the former case involves at most a 3 x 3 block---it can be mapped onto the time evolution of a time-independent Hamiltonian, so the dynamics can be evaluated analytic…
We study the bound states of $N$ identical bosons that are described by a multiband Bose-Hubbard model with generic hoppings and an attractive onsite interaction. Using a variational approach, we first derive exact integral equations for the dimers, trimers, tetramers, and other multimers, and then apply them to a one-dimensional sawtooth model that features two bands. In particular we reveal the presence of not only the offsite dimer states which consist of two monomers on different sites even in the strong-coupling limit but also the offsite trimer states which consist of either a dimer on o…
We find the eigenvalues $E_α $ and eigenvectors $|E_α>$ of the extended Hubbard dimer and we represent each part $E_α |E_α> < E_α |$ of the dimer Hamiltonian $(α=1,2,...,16)$in the second quantizations with the use of the Hubbard and spin operators. This procedure gives a review of all competitive, intrinsic interactions, deeply hidden in the orginal form of the considered Hamiltonian. Among competitive interactions we can find ferromagnetic and antiferromagnetic interactions (also in the form appearing in the $t-J$ model), hopping of the Cooper pairs between different dimer lattice sites (sim…
The constrained-search formulation of Levy and Lieb provides a concrete mapping from N-representable densities to the space of N-particle wavefunctions and explicitly defines the universal functional of density functional theory. We numerically implement the Levy-Lieb procedure for a paradigmatic lattice system, the Hubbard dimer, using a modified variational quantum eigensolver approach. We demonstrate density variational minimization using the resulting hybrid quantum-classical scheme featuring real-time computation of the Levy-Lieb functional along the search trajectory. We further illustra…
Exploiting the matrix-product-state based density-matrix renormalization group (DMRG) technique we study the one-dimensional extended ($U$-$V$) Hubbard model with explicit bond dimerization in the half-filled band sector. In particular we investigate the nature of the quantum phase transition, taking place with growing ratio $V/U$ between the symmetry-protected-topological and charge-density-wave insulating states. The (weak-coupling) critical line of continuous Ising transitions with central charge $c=1/2$ terminates at a tricritical point belonging to the universality class of the dilute Isi…
We present an unbiased numerical density-matrix renormalization group study of the one-dimensional Bose-Hubbard model supplemented by nearest-neighbor Coulomb interaction and bond dimerization. It places the emphasis on the determination of the ground-state phase diagram and shows that, besides dimerized Mott and density-wave insulating phases, an intermediate symmetry-protected topological Haldane insulator emerges at weak Coulomb interactions for filling factor one, which disappears, however, when the dimerization becomes too large. Analyzing the critical behavior of the model, we prove that…
In the most studied family of organic superconductors kappa-(BEDT-TTF)_2X, the BEDT-TTF molecules that make up the conducting planes are coupled as dimers. For some anions X, an antiferromagnetic insulator is found at low temperatures adjacent to superconductivity. With an average of one hole carrier per dimer, the BEDT-TTF band is effectively 1/2-filled. Numerous theories have suggested that fluctuations of the magnetic order can drive superconducting pairing in these models, even as direct calculations of superconducting pairing in monomer 1/2-filled band models find no superconductivity. He…
We consider a model Hamiltonian for a dimer including all the electronic one- and two-body terms consistent with a single orbital per site, a free Einstein phonon term, and an electron-phonon coupling of the Holstein type. The bare electronic interaction parameters were evaluated in terms of Wannier functions built from Gaussian atomic orbitals. An effective polaronic Hamiltonian was obtained by an unrestricted displaced-oscillator transformation, followed by evaluation of the phononic terms over a squeezed-phonon variational wave function. For the cases of quarter-filled and half-filled orbit…
A new class of orbital-dependent exchange-correlation (xc) potentials for applications in noncollinear spin-density-functional theory is developed. Starting from the optimized effective potential (OEP) formalism for the exact exchange potential - generalized to the noncollinear case - correlation effects are added via a self-consistent procedure inspired by the Singwi-Tosi-Land-Sjolander (STLS) method. The orbital-dependent xc potentials are applied to the Hubbard dimer in uniform and noncollinear magnetic fields and compared to exact diagonalization and to the Bethe-ansatz local spin-density …
Common density-matrix functionals, the Müller and the power functional, have been benchmarked for the half-filled Hubbard dimer, which allows to model the bond dissociation problem and the transition from the weakly to the strongly correlated limit. Unbiased numerical calculations are combined with analytical results. Despite the well known successes of the Müller functional, the ground state is degenerate with a one-dimensional manifold of ferromagnetic solutions. The resulting infinite magnetic susceptibility indicates another qualitative flaw of Müller's functional. The derivative discontin…
We explore the quantum phase transition between Peierls and charge-density-wave insulating states in the one-dimensional, half-filled, extended Hubbard model with explicit bond dimerization. We show that the critical line of the continuous Ising transition terminates at a tricritical point, belonging to the universality class of the tricritical Ising model with central charge $c=7/10$. Above this point, the quantum phase transition becomes first order. Employing a numerical matrix-product-state based (infinite) density-matrix renormalization group method we determine the ground-state phase dia…
Superconductivity in the Hubbard model is studied on a series of lattices in which dimers are coupled in various types of arrays. Using fluctuation exchange method and solving the linearized Eliashberg equation, the transition temperature $T_c$ of these systems is estimated to be much higher than that of the Hubbard model on a simple square lattice, which is a model for the high $T_c$ cuprates. We conclude that these `dimer array' systems can generally exhibit superconductivity with very high $T_c$. Not only $d$-electron systems, but also $p$-electron systems may provide various stages for rea…
The formation of a spin glass usually requires both structural disorder and frustrated magnetic interactions. Consequently, the origin of spin-glass behaviour in Y$_2$Mo$_2$O$_7$ $-$ in which magnetic Mo$^{4+}$ ions occupy a frustrated pyrochlore lattice with minimal compositional disorder $-$ has been a longstanding question. Here, we use neutron and X-ray pair-distribution function (PDF) analysis to develop a disorder model that resolves apparent incompatibilities between previously-reported PDF, EXAFS and NMR studies and provides a new and physical mechanism for spin-glass formation. We sho…
In this paper we revisit the glass model describing the macroscopic behavior of the High-Temperature superconductors. We link the glass model at the microscopic level to the striped phase phenomenon, recently discussed widely. The size of the striped phase domains is consistent with earlier predictions of the glass model when it was introduced for High-Temperature Superconductivity in 1987. In an additional step we use the Hubbard model to describe the microscopic mechanism for d-wave pairing within these finite size stripes. We discuss the implications for superconducting correlations of Hubb…
The ground state of a one-dimensional Hubbard model with a bond-charge attraction W term at half-filling is investigated by the density matrix renormalization group method. It is confirmed that the spin gap will be closed at U>8W. But the long-range bond order wave survives even when the spin gap is closed. It indicates that the ground state is a novel dimerized spin fluid at U>8W. By a charge-spin transformation, it is shown that there should be a dimerized metallic phase at U<-8W. Furthermore, it is found that the Hubbard interaction U enhances initially the dimerization for a weak bond char…
The asymmetric Hubbard dimer is used to study the density-dependence of the exact frequency-dependent kernel of linear-response time-dependent density functional theory. The exact form of the kernel is given, and the limitations of the adiabatic approximation utilizing the exact ground-state functional are shown. The oscillator strength sum rule is proven for lattice Hamiltonians, and relative oscillator strengths are defined appropriately. The method of Casida for extracting oscillator strengths from a frequency-dependent kernel is demonstrated to yield the exact result with this kernel. An u…
The competition between evolution time, interaction strength, and temperature challenges our understanding of many-body quantum systems out-of-equilibrium. Here we consider a benchmark system, the Hubbard dimer, which allows us to explore all the relevant regimes and calculate exactly the related average quantum work. At difference with previous studies, we focus on the effect of increasing temperature, and show how this can turn competition between many-body interactions and driving field into synergy. We then turn to use recently proposed protocols inspired by density functional theory to ex…
We study relaxation dynamics in a strongly-interacting two-site Fermi-Hubbard model that is induced by coupling each site to a local fermionic bath. To derive the proper form of the Lindblad operators that enter an effective description of the system-bath coupling in different temperature regimes, we employ a diagrammatic real-time technique for the time evolution of the reduced density matrix. In spite of a local coupling to the baths, the found Lindblad operators are non-local in space. We compare with the local approximation, where those non-local effects are neglected. Furthermore, we prop…
Gross-Oliveira-Kohn density-functional theory (GOK-DFT) is an extension of DFT to excited states where the basic variable is the ensemble density, i.e. the weighted sum of ground- and excited-state densities. The ensemble energy (i.e. the weighted sum of ground- and excited-state energies) can be obtained variationally as a functional of the ensemble density. Like in DFT, the key ingredient to model in GOK-DFT is the exchange-correlation functional. Developing density-functional approximations (DFAs) for ensembles is a complicated task as both density and weight dependencies should in principl…
The Bethe-Salpeter equation (BSE) is the key equation in many-body perturbation theory based on Green's functions to access response properties. Within the $GW$ approximation to the exchange-correlation kernel, the BSE has been successfully applied to several finite and infinite systems. However, it also shows some failures, such as underestimated triplet excitation energies, lack of double excitations, ground-state energy instabilities in the dissociation limit, etc. In this work, we study the performance of the BSE within the $GW$ approximation as well as the $T$-matrix approximation for the…
Using quantum Monte Carlo simulations, results of a strong-coupling expansion, and Luttinger liquid theory, we determine quantitatively the ground state phase diagram of the one-dimensional extended Hubbard model with on-site and nearest-neighbor repulsions U and V. We show that spin frustration stabilizes a bond-ordered (dimerized) state for U appr. V/2 up to U/t appr. 9, where t is the nearest-neighbor hopping. The transition from the dimerized state to the staggered charge-density-wave state for large V/U is continuous for U up to appr. 5.5 and first-order for higher U.
Using the simple (symmetric) Hubbard dimer, we analyze some important features of the $GW$ approximation. We show that the problem of the existence of multiple quasiparticle solutions in the (perturbative) one-shot $GW$ method and its partially self-consistent version is solved by full self-consistency. We also analyze the neutral excitation spectrum using the Bethe-Salpeter equation (BSE) formalism within the standard $GW$ approximation and find, in particular, that i) some neutral excitation energies become complex when the electron-electron interaction $U$ increases, which can be traced bac…
Most realistic calculations of moderately correlated materials begin with a ground-state density functional theory (DFT) calculation. While Kohn-Sham DFT is used in about 40,000 scientific papers each year, the fundamental underpinnings are not widely appreciated. In this chapter, we analyze the inherent characteristics of DFT in their simplest form, using the asymmetric Hubbard dimer as an illustrative model. We begin by working through the core tenets of DFT, explaining what the exact ground-state density functional yields and does not yield. Given the relative simplicity of the system, almo…
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.
We investigate the microscopic dynamics at the initial stage of photoinduced phase transitions in tetrathiafulvalene-$p$-chloranil by exact diagonalization. We first show that the one-dimensional extended Peierls-Hubbard model exhibits a neutral phase with small ionicity and negligible dimerization and an ionic phase with moderate ionicity and dimerization. Aside from these phases, we find a doubly ionized phase with strong dimerization that we call the "dipole" phase. These ground-state phases are characterized by various order parameters and the Zak phase, which is relevant to electronic pol…
We calculate the effective Coulomb interactions between holes in dimers of the organic molecule BEDT-TTF in vacuo. We use density functional theory (DFT) to parameterise Hubbard models for beta and kappa phase organic charge transfer salts. We focus on the intra-dimer Coulomb repulsion, U_d, and the inter-monomer Coulomb repulsion, V_m. We find that U_d = 3.22 \pm 0.09 eV and V_m = 2.71 \pm 0.10 eV for 23 experimental geometries taken from a range of materials in both polymorphs. The quoted error is one standard deviation over the set of conformations studied. We conclude that U_d and V_m are …
The hardcore extended Bose Hubbard model on a bilayer square lattice with attractive (repulsive) interplane (intraplane) interactions has been investigated numerically. Focusing on the strong interplane hopping parameter regime, triplet valence bonds of dimers are stabilized to form valence bond checkerboard solid at quarter filling. Increasing the particle number we confirm, for the first time, the presence of the exotic valence bond supersolid phase, where the valence bond solid ordering and the boson superfluidity coexist. Rising further the particle number will lead to a checkerboard solid…
We study the charge ordering (CO) in the one-dimensional (1D) extended Hubbard model at quarter filling where the nearest-neighbor Coulomb repulsion and dimerization in the hopping parameters are included. Using the cluster mean-field approximation to take into account the effect of quantum fluctuations, we determine the CO phase boundary of the model in the parameter space at T=0 K. We thus find that the dimerization suppresses the stability of the CO phase strongly, and in consequence, the realistic parameter values for quasi-1D organic materials such as (TMTTF)$_2$PF$_6$ are outside the reg…
We study the one-dimensional quarter-filled extended Hubbard model with an alternating transfer integral. In the strong-dimerization limit the charge part is described by the quantum Ising model which shows the two-dimensional Ising criticality at the self-dual point, and it is naturally connected to the double-frequency sine-Gordon theory in the weak dimerization. Treating low-lying excitations in finite-size systems, we numerically determine a phase boundary between two types of $4k_{\rm F}$ density-wave states and clarify the ground-state phase diagram. Further, we refer to its relevances t…
We study the problem of designing an artificial Mott insulator in a correlated oxide heterostructure. We consider the extreme limit of quantum confinement based on ionic discontinuity doping, and argue that a unique dimer Mott insulator can be achieved for the case of a single SrO layer in a GdTiO$_3$ matrix. In the dimer Mott insulator, electrons are localized not to individual atoms but to bonding orbitals on molecular dimers formed across a bilayer of two TiO$_2$ planes, and is analogous to the Mott insulating state of Hubbard ladders, studied in the 1990s. We verify the existence of the di…
We consider a symplectic extrapolation of the Hubbard model of N fold replicated electrons and solve this model exactly in two special cases, at N=infinity in the bosonic sector and for any N on a dimer of two points. At N=infinity we find a multiplet of collective modes that contains neutral spin fluctuations and charged pair fluctuations that are degenerate with each other at zero doping. Our solution of the symplectic model on a dimer of two points for any N interpolates smoothly between N=1 and N=infinity without any visible discontinuity. These results suggest that the inclusion of charge…
We present a Monte Carlo study of the Bose-glass to superfluid transition in the three-dimensional Bose-Hubbard model. Simulations are performed on the classical (3 + 1) dimensional link-current representation using the geometrical worm algorithm. Finite-size scaling analysis (on lattices as large as 16x16x16x512 sites) of the superfluid stiffness and the compressibility is consistent with a value of the dynamical critical exponent z = 3, in agreement with existing scaling and renormalization group arguments that z = d. We find also a value of $ν= 0.70(12)$ for the correlation length exponent,…
This review explains the relationship between density functional theory and strongly correlated models using the simplest possible example, the two-site Hubbard model. The relationship to traditional quantum chemistry is included. Even in this elementary example, where the exact ground-state energy and site occupations can be found analytically, there is much to be explained in terms of the underlying logic and aims of Density Functional Theory. Although the usual solution is analytic, the density functional is given only implicitly. We overcome this difficulty using the Levy-Lieb construction…
We investigate ground state properties of the half-filled staggered-flux Hubbard model on a square lattice. Energy gaps to charge and spin excitations and magnetic as well as dimer orders are calculated as a function of interaction strength $U/t$ by means of constrained-path quantum Monte Carlo method. It is found that the system is a semi-metal at $U/t\lesssim 5.6$ and a Mott insulator with long-range antiferromagnetic order at $U/t \gtrsim 6.6$. In the range $5.6\lesssim U/t\lesssim 6.6$, the ground state is an correlated insulator where both magnetic and dimer orders are absent. Furthermore…
We consider the one-dimensional extended Hubbard model in the presence of an explicit dimerization $δ$. For a sufficiently strong nearest neighbour repulsion we establish the existence of a quantum phase transition between a mixed bond-order wave and charge-density wave phase from a pure bond-order wave phase. This phase transition is in the universality class of the two-dimensional Ising model.
Expansion dynamics of interacting fermions in a lattice are simulated within the one-dimensional (1D) Hubbard model, using the essentially exact time-evolving block decimation (TEBD) method. In particular, the expansion of an initial band-insulator state is considered. We analyze the simulation results based on the dynamics of a two-site two-particle system, the so-called Hubbard dimer. Our findings describe essential features of a recent experiment on the expansion of a Fermi gas in a two-dimensional lattice. We show that the Hubbard-dimer dynamics, combined with a two-fluid model for the pai…
The molar spin susceptibilities $χ(T)$ of Na-TCNQ, K-TCNQ and Rb-TCNQ(II) are fit quantitatively to 450 K in terms of half-filled bands of three one-dimensional Hubbard models with extended interactions using exact results for finite systems. All three models have bond order wave (BOW) and charge density wave (CDW) phases with boundary $V = V_c(U)$ for nearest-neighbor interaction $V$ and on-site repulsion $U$. At high $T$, all three salts have regular stacks of $\rm TCNQ^-$ anion radicals. The $χ(T)$ fits place Na and K in the CDW phase and Rb(II) in the BOW phase with $V \approx V_c$. The Na…
