Hubbard-Holstein Model Research Articles

Variational Lang-Firsov Approach Plus Møller-Plesset Perturbation Theory with Applications to Ab Initio Polariton Chemistry.

We apply the Lang-Firsov (LF) transformation to electron-boson coupled Hamiltonians and variationally optimize the transformation parameters and molecular orbital coefficients to determine the ground state. Møller-Plesset (MP-n, with n = 2 and 4) perturbation theory is then applied on top of the optimized LF mean-field state to improve the description of electron-electron and electron-boson correlations. The method (LF-MP) is applied to several electron-boson coupled systems, including the Hubbard-Holstein model, diatomic molecule dissociation (H2, HF), and the modification of proton transfer reactions (malonaldehyde and aminopropenal) via the formation of polaritons in an optical cavity. We show that with a correction for the electron-electron correlation, the method gives quantitatively accurate energies comparable to that by exact diagonalization or coupled-cluster theory. The effects of multiple photon modes, spin polarization, and the comparison to the coherent state MP theory are also discussed.

A semi-exact analytical study of the phase diagram of the two dimensional extended Holstein-Hubbard model

An extended Holstein-Hubbard model is studied in two dimensions analytically for both weak and strong correlations. A series of unitary transformations followed by an averaging with a fully-generalized phonon state is performed to obtain an effective electronic Hamiltonian. For weak correlation, the Hartree-Fock approximation is employed and for strong correlation, the electronic Hamiltonian is mapped on to an effective t−J model which is solved using the Gutzwiller approximation and the Zubarev technique. The variation of the effective Coulomb interaction strength is studied for different electron-electron and electron-phonon interaction coefficients and the SDW-CDW phase diagram is plotted using the Mott-Hubbard metallicity criteria. A metallic regime is shown to exist at the SDW-CDW cross-over region and the width of the metallic phase is found to be broader than the result obtained by numerical diagonalization method. The metallic phase is also found to be wider in two dimensions than in one dimension.

A hybrid quantum-classical method for electron-phonon systems

Interactions between electrons and phonons play a crucial role in quantum materials. Yet, there is no universal method that would simultaneously accurately account for strong electron-phonon interactions and electronic correlations. By combining methods of the variational quantum eigensolver and the variational non-Gaussian solver, we develop a hybrid quantum-classical algorithm suitable for this type of correlated systems. This hybrid method tackles systems with arbitrarily strong electron-phonon coupling without increasing the number of required qubits and quantum gates, as compared to purely electronic models. We benchmark our method by applying it to the paradigmatic Hubbard-Holstein model at half filling, and show that it correctly captures the competition between charge density wave and antiferromagnetic phases, quantitatively consistent with exact diagonalization.

Dynamical mean-field theory for the Hubbard-Holstein model on a quantum device

Utilizing quantum computers for real materials calculations is highly promising, but hinges on qubit fidelity and fault-tolerant quantum algorithms. Here, the authors show that a representative model for real materials, the Hubbard-Holstein model, which describes interacting electrons on a lattice coupled to bosonic degrees of freedom, can be reliably investigated using today's Quantum processors. Adapting state-of-the-art mathematical tools is shown to be critical to overcome limitations imposed by the noise level of current quantum computing devices.

Self-trapping transition in a two dimensional extended Holstein-Hubbard model: A mean-field study

The self-trapping transition is studied within the framework of the two-dimensional extended Holstein-Hubbard model for both adiabatic and anti-adiabatic cases. A highly accurate phonon state is chosen as the averaging state to obtain an effective electronic Hamiltonian which is solved for the two different Coulomb correlation regimes separately. For weak correlation, the Hartree-Fock mean-field approximation is employed and for strong correlation, the electronic Hamiltonian is mapped on to an effective t−J model which is solved by using the Gutzwiller approximation and the Zubarev Green's function technique. For the entire range of the Coulomb interaction, the self-trapping transition turns out to be continuous for the anti-adiabatic regime and for the adiabatic regime the transition it is found to be discontinuous.

Chebyshev pseudosite matrix product state approach for the spectral functions of electron-phonon coupling systems

The electron-phonon ($e$-ph) coupling system often has a large number of phonon degrees of freedom, whose spectral functions are numerically difficult to compute using matrix product state (MPS) formalisms. To solve this problem, we propose a new and practical method that combines the Chebyshev MPS and the pseudosite density matrix renormalization group (DMRG) algorithm. The Chebyshev vector is represented by a pseudosite MPS with global $U(1)$ fermion symmetry, which maps $2^{N_p}$ bosonic degrees of freedom onto $N_p$ pseudosites, each with two states. This approach can handle arbitrary $e$-ph coupling Hamiltonians where pseudosite DMRG performs efficiently. We use this method to study the spectral functions of the doped extended Hubbard-Holstein model in a regime of strong Coulomb repulsion, which has not been studied extensively before. Key features of the excitation spectra are captured at a modest computational cost. Our results show that weak extended $e$-ph couplings can increase the spectral weight of the holon-folding branch at low phonon frequencies, in agreement with angle-resolved photoemission observations on one-dimensional (1D) cuprates.

Quantum simulation of extended electron-phonon-coupling models in a hybrid Rydberg atom setup

State-of-the-art experiments using Rydberg atoms can now operate with large numbers of trapped particles with tunable geometry and long coherence time. We propose a way to utilize this in a hybrid setup involving neutral ground-state atoms to efficiently simulate condensed-matter models featuring electron-phonon coupling. Such implementation should allow for controlling the coupling strength and range as well as the band structure of both the phonons and atoms, paving the way towards studying both static and dynamic properties of extended Hubbard-Holstein models.

Magnetic impurities in a charge-ordered background

We investigate how magnetic impurities may affect a system exhibiting a charge-density wave (CDW) in its ground state. We consider a disordered Hubbard-Holstein model with a homogeneous electron-phonon interaction but with a (randomly chosen) fraction of sites displaying a nonzero Coulomb repulsion $U$ and perform state-of-the-art finite-temperature quantum Monte Carlo simulations. For a single magnetic impurity, charge-charge correlations hamper the spin-spin ones around the repulsive site, thus requiring a strong enough value of $U$ to create nonnegligible antiferromagnetic (AFM) correlations. As the number of magnetic impurities increases, these AFM correlations become deleterious to CDW order and its features. First, the critical temperature is drastically reduced and seems to vanish $\ensuremath{\sim}40%$ of impurities (for fixed $U/\ensuremath{\lambda}=2$), which we correlate with the classical percolation threshold. We also notice that just a small amount of disorder suffices to create a bad insulating state, with the suppression of both Peierls and spin gaps, even within the charge-ordered phase. Finally, we have also found that pairing correlations are enhanced at large doping, driven by the competition between CDW and AFM tendencies.

Nonequilibrium spectral moment sum rules of the Holstein–Hubbard model

We derive a general procedure for evaluating the nth derivative of a time-dependent operator in the Heisenberg representation and employ this approach to calculate the zeroth to third spectral moment sum rules of the retarded electronic Green’s function and self-energy for a system described by the Holstein–Hubbard model allowing for arbitrary spatial and time variation of all parameters (including spatially homogeneous electric fields and parameter quenches). For a translationally invariant (but time-dependent) Hamiltonian, we also provide sum rules in momentum space. The sum rules can be applied to various different phenomena like time-resolved angle-resolved photoemission spectroscopy and benchmarking the accuracy of numerical many-body calculations. This work also corrects some errors found in earlier work on simpler models.

Stripe correlations in the two-dimensional Hubbard-Holstein model

Several state-of-the-art numerical methods have observed static or fluctuating spin and charge stripes in doped two-dimensional Hubbard models, suggesting that these orders play a significant role in shaping the cuprate phase diagram. Many experiments, however, also indicate that the cuprates have strong electron-phonon (e-ph) coupling, and it is unclear how this interaction influences stripe correlations. We study static and fluctuating stripe orders in the doped single-band Hubbard-Holstein model using zero temperature variational Monte Carlo and finite temperature determinant quantum Monte Carlo. We find that the lattice couples more strongly with the charge component of the stripes, leading to an enhancement or suppression of stripe correlations, depending on model parameters like the next-nearest-neighbor hopping {t}^{{prime} } or phonon energy Ω. Our results help elucidate how the e-ph interaction can tip the delicate balance between stripe and superconducting correlations in the Hubbard-Holstein model with implications for our understanding of the high-Tc cuprates.

On the uniaxial strain (pressure) derivatives of the critical temperature of superconductivity of La[formula omitted]Sr[formula omitted]CuO4

The uniaxial strain (and pressure) derivatives of the critical temperature of superconductivity (Tc) of La2−xSrxCuO4 (LSCO) cuprate is studied within the extended Holstein–Hubbard model and bipolaronic mechanism of high-Tc superconductivity. The Tc of LSCO cuprate is associated with the Bose–Einstein condensation temperature, TBEC, of an ideal gas of the intersite bipolarons. Working on a modified chain model lattice of cuprates we expressed TBEC as a function of LSCO’s parameters (apical oxygen ion’s mass and its vibration frequency, the periods and the strains of the lattice). Using the experimental values of LSCO parameters the uniaxial strain derivatives and the uniaxial pressure derivatives of Tc i.e. ∂TBEC/∂ɛi and ∂TBEC/∂pi (i=a,c) of LSCO cuprate are calculated as a function of doping level x. The obtained results are compared with the available experimental data. Qualitative and quantitative similarities as well as differences in the doping dependencies of between our ∂TBEC/∂ɛi (∂TBEC/∂pi) and the experimental ∂Tc/∂ɛi (∂Tc/∂pi) (i=a,c) are discussed. Upon carried analysis we concluded that the uniaxial strain (pressure) derivatives of Tc of LSCO cuprate can be interpreted within the bipolaronic mechanism of superconductivity.

Provably accurate simulation of gauge theories and bosonic systems

Quantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit Λ on each local quantum number, and if the initial state has low local quantum numbers, then an error at most ϵ can be achieved by choosing Λ to scale polylogarithmically with ϵ−1, an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on Λ that achieves accuracy ϵ, obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error in time evolution, we also prove that spectrally isolated energy eigenstates can be approximated with accuracy ϵ by truncating local quantum numbers at Λ=polylog(ϵ−1).

A semi-exact study of self-trapping transition in a one-dimensional Holstein-Hubbard model

The nature of self-trapping transition in the one-dimensional extended Holstein-Hubbard model is investigated for both adiabatic and anti-adiabatic regimes. The coherence and correlations of the phonons are incorporated accurately by applying a sequence of unitary transformations and a fully-generalized many-phonon wave function is chosen as the averaging phonon state to obtain an effective electronic problem. The renormalized electronic problem is subsequently treated exactly using the method of Bethe ansatz and the self-trapping transition is shown to be continuous over the entire range of the material parameters.

Time-linear scaling nonequilibrium Green's function method for real-time simulations of interacting electrons and bosons. II. Dynamics of polarons and doublons

Nonequilibrium dynamics of the open chain Holstein-Hubbard model is studied using the linear time-scaling GKBA+ODE scheme developed in the preceeding paper. We focus on the set of parameters relevant for photovoltaic materials, i.\,e., a pair of electrons interacting with phonons at the cross-over between the adiabatic and anti-adiabatic regimes and at moderately large electron-electron interaction. By comparing with exact solutions for two corner cases, we demonstrate the accuracy of the $T$-matrix (in the $pp$ channel) and the second-order Fan ($GD$) approximations for the treatment of electronic ($e$-$e$) and electron-phonon ($e$-ph) correlations, respectively. The feedback of electron on phonons is consistently included and is shown to be mandatory for the total energy conservation. When two interactions are simultaneously present, our simulations offer a glimpse into the dynamics of doublons and polarons unveiling the formation, propagation and decay of these quasiparticles, energy redistribution between them and self-trapping of electrons.

Effect of the hole concentration on the polaronic mobility for the strongly interacting electrons

The effective hopping of the polaron has been studied for different hole concentrations. Holstein-Hubbard model is used for the half-filled and less than half-filled cases. A series of canonical transformations are imposed on the generic Hamiltonian and considering the strongly interacting electrons, the transformed Hamiltonian has been reconstructed to an effective t-J model. Using the mean-filled Hartree-Fock approximation the electronic Hamiltonian has been solved with the Zuberev’s Green function technique. The effective polaronic transport is found to be strongly dependent on the electron and hole concentration of the system.

Phase diagram of the three-orbital Hubbard-Holstein model simulating Superconducting Tungsten Bronze AxWO3

The superconductivity in tungsten bronze Ax WO3 (A═alkali metal) is studied based on first-principles calculations and orbital fluctuation theory. We discuss the effects of the electron-Jahn-Teller phonon interaction and the on-site Coulomb interaction in the random phase approximation, and obtain a phase diagram on the U — g plane in the low-doped regime at x~0.05, where high-temperature superconductivity has been experimentally observed.

Many-Body Quantum State Diffusion for Non-Markovian Dynamics in Strongly Interacting Systems.

Capturing non-Markovian dynamics of open quantum systems is generally a challenging problem, especially for strongly interacting many-body systems. In this Letter, we combine recently developed non-Markovian quantum state diffusion techniques with tensor network methods to address this challenge. As a first example, we explore a Hubbard-Holstein model with dissipative phonon modes, where this new approach allows us to quantitatively assess how correlations spread in the presence of non-Markovian dissipation in a 1D many-body system. We find regimes where correlation growth can be enhanced by these effects, offering new routes for dissipatively enhancing transport and correlation spreading, relevant for both solid state and cold atom experiments.

Pair-density-wave in the strong coupling limit of the Holstein-Hubbard model

A pair-density-wave (PDW) is a superconducting state with an oscillating order parameter. A microscopic mechanism that can give rise to it has been long sought but has not yet been established by any controlled calculation. Here we report a density-matrix renormalization-group (DMRG) study of an effective t-J-V model, which is equivalent to the Holstein-Hubbard model in a strong-coupling limit, on long two-, four-, and six-leg triangular cylinders. While a state with long-range PDW order is precluded in one dimension, we find strong quasi-long-range PDW order with a divergent PDW susceptibility as well as the spontaneous breaking of time-reversal and inversion symmetries. Despite the strong interactions, the underlying Fermi surfaces and electron pockets around the K and {K}^{prime} points in the Brillouin zone can be identified. We conclude that the state is valley-polarized and that the PDW arises from intra-pocket pairing with an incommensurate center of mass momentum. In the two-leg case, the exponential decay of spin correlations and the measured central charge c ≈ 1 are consistent with an unusual realization of a Luther-Emery liquid.

Strength of the Hubbard potential and its modification by breathing distortion in BaBiO3

${\text{BaBiO}}_{3}$ compound is known as an archetype example of a three-dimensional Holstein model with the realization of the charge-density wave state at half filling and the superconducting state when doped. Although many works are devoted to the study the electron--phonon interaction in ${\text{BaBiO}}_{3}$, the influence of the electron--electron Hubbard interaction on the electronic structure in this system is still under investigation. In our work, we obtain analytical expression for the screened Coulomb potential, and along with the basis of ab initio-computed maximally localized Wannier orbitals, we quantitatively estimate the magnitude of the effective on-site Hubbard potential scrutinizing the effects of distortion of the crystal lattice. We show that a proper inclusion of the electron--electron interactions into the Holstein model significantly lowers the value of the underlying electron--phonon coupling. Finally, we find that the amplitudes of the repulsive electron--electron potential and its attractive counterpart mediated by the electron--phonon coupling are rather comparable. This may open a way for a realization of the intermediate phase of ${\text{BaBiO}}_{3}$ in terms of the Holstein-Hubbard model.

Strong Correlation Effects and Localization in Metallic Systems

We have studied strong correlation effects and localization in metallic systems. Strong correlation effects and localization occurred in metallic systems due to strong electron-electron interactions and strong electron phonon coupling. Strong electron-phonon interactions have been found in such materials as cuprates, fullerides and manganites. The interplay of electronelectron and electron-phonon interactions in these correlated systems leaded to coexistence of or competition between various phases such as super conductivity, charge-density-wave or spin-density wave phases or formation of novel non Fermi liquid phases, polarons, bipolarons and so on. We have considered Holstein-Hubbard model for our work. The Holstein-Hubbard model provides on over simplified description of both electron-electron and electron-phonon interactions, it retains to be the relevant ingredients of a system in which electrons experience simultaneously an instantaneous short range repulsion and a phononmediated retarded attraction. In spite of its formal simplicity, it is not exactly solvable even in one dimension. So quantum simulators of Fermi-Hubbard modes based on either fermionic atoms inoptical lattice or electrons in artificial quantum dot crystals have been used. In our work a classical analog simulator was theoretically proposed for the two site Holstein-Hubbard model based on light transport in engineered waveguide lattices, which is capable of reproducing the temporal dynamics of the quantum model in Fock space as spatial evolution in photonic lattice. We have found that in the strong correlation regime the periodic temporal dynamics exhibited by the Holstein-Hubbard Hamiltonian, related to the excitation Holstein polarons has been explained in terms of generalized Bloch oscillations of a single particle in a semi-infinite inhomogeneous tight binding lattice. Light transport in two dimensional photonic lattices simulated in Fock space the hopping dynamics of two correlated electrons on a one dimensional lattice and exploited to visualize such phenomena as correlated tunneling of bond electron-electron molecules and their coherent motion under the action of d.c. or a.c. fields. The obtained results were found in good agreement with previous results.