Spinless Holstein Model Research Articles

Electron addition spectral functions of low-density polaron liquids

Spectral functions are important quantities that contain a wealth of information about the quasiparticles of a system, and that can also be measured experimentally. For systems with electron-phonon coupling, good approximations for the spectral function are available only in the Migdal limit (at Fermi energies much larger than the typical phonon frequency, E_F\gg \OmegaEF≫Ω, requiring a large carrier concentration xx) and in the single polaron limit (at x=0x=0). Here we show that the region with x\ll 1x≪1(E_F<\OmegaEF<Ω) can also be reliably investigated with the Momentum Average (MA) variational approximation, which essentially describes the formation of a polaron above an inert Fermi sea. Specifically, we show that for the one-dimensional spinless Holstein model, the MA spectral functions compare favorably with those calculated using variationally exact density matrix renormalization group simulations (DMRG) evaluated directly in frequency-space, so long as x<0.1x<0.1 and the adiabaticity ratio \Omega/t>0.5Ω/t>0.5. Unlike in the Migdal limit, here 'polaronic physics’ emerges already at moderate couplings. The relevance of these results for a spinful low-xx metal is also discussed.

Density-matrix renormalization group study of the linear conductance in quantum wires coupled to interacting leads or phonons

In a previous paper [J.-M. Bischoff and E. Jeckelmann, Phys. Rev. B 96, 195111 (2017)] we introduced a density-matrix renormalization group method for calculating the linear conductance of one-dimensional correlated quantum systems and demonstrated it on homogeneous spinless fermion chains with impurities. Here we present extensions of this method to inhomogeneous systems, models with phonons, and the spin conductance of electronic models. The method is applied to a spinless fermion wire-lead model, the homogeneous spinless Holstein model, and the Hubbard model. Its capabilities are demonstrated by comparison with the predictions of Luttinger liquid theory combined with Bethe Ansatz solutions and other numerical methods. We find a complex behavior for quantum wires coupled to interacting leads when the sign of the interaction (repulsive/attractive) differs in wire and leads. The renormalization of the conductance given by the Luttinger parameter in purely fermionic systems is shown to remain valid in the Luttinger liquid phase of the Holstein model with phononic degrees of freedom.

Langevin approach to lattice dynamics in a charge-ordered polaronic system

We use a Langevin approach to treat the finite temperature dynamics of displacement variables in the half-filled spinless Holstein model. Working in the adiabatic regime we exploit the smallness of the adiabatic parameter to simplify the memory effects and estimate displacement costs from an "instantaneous" electronic Hamiltonian. We use a phenomenological damping rate, and uncorrelated thermal noise. The low temperature state has checkerboard charge order (CO) and the Langevin scheme generates equilibrium thermodynamic properties that accurately match Monte Carlo results. It additionally yields the dynamical structure factor, $D({\bf q}, \omega)$, from the displacement field $x({\bf r}, t)$. We observe four regimes with increasing temperature, $T$, classified in relation to the charge ordering temperature, $T_c$, and the `polaron formation' temperature $T_P$, with $ T_c \ll T_P$. For $T \ll T_c$ the oscillations are harmonic, leading to dispersive phonons, with increasing $T$ bringing in anharmonic, momentum dependent, corrections. For $T \sim T_c$, thermal tunneling events of the $x({\bf r})$ field occur, with a propagating `domain' pattern at wavevector ${\bf q} \sim (\pi, \pi)$ and low energy weight in $D({\bf q}, \omega)$. When $T_c < T < T_P$, the disordered polaron regime, domain structures vanish, the dispersion narrows, and low energy weight is lost. For $T \gtrsim T_P$ we essentially have uncorrelated local oscillations. We propose simple models to analyse this rich dynamics.

Thermal and quantum lattice fluctuations in Peierls chains

The thermodynamic and spectral properties of electrons coupled to quantum phonons are studied within the spinless Holstein model. Using quantum Monte Carlo simulations, we obtain accurate results for the specific heat and the compressibility, covering the entire range of electron-phonon couplings and phonon frequencies. To this end, we derive an efficient estimator for the specific heat using the properties of the perturbation expansion. This allows us to quantitatively test the predictions of Tomonaga-Luttinger liquid theory as well as the widely used adiabatic approximation for low phonon frequencies. A comparison with the spectral functions of electrons and phonons reveals that the formation of polaron excitations as well as the renormalization of the phonon mode across the Peierls transition have a pronounced effect on the specific heat in the adiabatic regime.

Dynamic charge correlations near the Peierls transition

The quantum phase transition between a repulsive Luttinger liquid and an insulating Peierls state is studied in the framework of the one-dimensional spinless Holstein model. We focus on the adiabatic regime but include the full quantum dynamics of the phonons. Using continuous-time quantum Monte Carlo simulations, we track in particular the dynamic charge structure factor and the single-particle spectrum across the transition. With increasing electron-phonon coupling, the dynamic charge structure factor reveals the emergence of a charge gap, and a clear signature of phonon softening at the zone boundary. The single-particle spectral function evolves continuously across the transition. Hybridization of the charge and phonon modes of the Luttinger liquid description leads to two modes, one of which corresponds to the coherent polaron band. This band acquires a gap upon entering the Peierls phase, whereas the other mode constitutes the incoherent, high-energy spectrum with backfolded shadow bands. Coherent polaronic motion is a direct consequence of quantum lattice fluctuations. In the strong-coupling regime, the spectrum is described by the static, mean-field limit. Importantly, whereas finite electron density in general leads to screening of polaron effects, the latter reappear at half filling due to charge ordering and lattice dimerization.

Formation and temperature evolution of correlated polarons in colossal magnetoresistivemanganites

We investigate the temperature dependence of charge ordered clusters in manganites. Interms of an approximate treatment of the half-filled spinless Holstein model incorporatingthe double exchange effect and nearest-neighbor Coulomb interactions, we find that chargeordering and polaronic states appear simultaneously which naturally defines a correlatedpolaron state. Without the double exchange, the temperature evolution of the charge(polaron) ordering parameter exhibits a broad peak structure only in the stronglycorrelated quasiadiabatic regime. Including the double exchange, the peak becomessharper and is observed both in the quasiadiabatic and antiadiabatic regimes.The peak structure in the quasiadiabatic regime provides an explanation for thetemperature dependence of the polaron correlation in nanoscale charge ordered polaronclusters observed in the manganites with intermediate to narrow bandwidths.

Carrier-density effects in many-polaron systems

Many-polaron systems with finite charge-carrier density are often encounteredexperimentally. However, until recently, no satisfactory theoretical description of thesesystems has been available even in the framework of simple models such as theone-dimensional spinless Holstein model considered here. In this work, previous resultsobtained using numerical as well as analytical approaches are reviewed from a unifiedperspective, focusing on spectral properties which reveal the nature of the quasiparticles inthe system. In the adiabatic regime and for intermediate electron–phonon coupling, acarrier-density driven crossover from a polaronic to a rather metallic system takes place.Further insight into the effects due to changes in density is gained by calculating thephonon spectral function, and the fermion–fermion and fermion–lattice correlationfunctions. Finally, we provide strong evidence against the possibility of phase separation.

Optical conductivity of polaronic charge carriers

The optical conductivity of charge carriers coupled to quantum phonons is studied in theframework of the one-dimensional spinless Holstein model. For one electron, variationaldiagonalization yields exact results in the thermodynamic limit, whereas at finite carrierdensity analytical approximations based on previous work on single-particle spectralfunctions are obtained. Particular emphasis is put on deviations from weak-coupling,small-polaron or one-electron theories occurring at intermediate coupling and/or finitecarrier density. The analytical results are in surprisingly good agreement with exact data,and exhibit the characteristic polaronic excitations observed in experiments on manganites.

Analytical approach to the quantum-phase transition in the one-dimensional spinless Holstein model

We study the one-dimensional Holstein model of spinless fermions interacting with dispersion-less phonons by using a recently developed projector-based renormalization method (PRM). At half-filling the system shows a metal-insulator transition to a Peierls distorted state at a critical electron-phonon coupling where both phases are described within the same theoretical framework. The transition is accompanied by a phonon softening at the Brillouin zone boundary and a gap in the electronic spectrum. For different filling, the phonon softening appears away from the Brillouin zone boundary and thus reflects a different type of broken symmetry state.

Spectral functions of the spinless Holstein model

An analytical approach to the one-dimensional spinless Holstein model is proposed, whichis valid at finite charge-carrier concentrations. Spectral functions of charge carriers arecomputed on the basis of self-energy calculations. A generalization of the Lang–Firsovcanonical transformation method is shown to provide an interpolation scheme between theextreme weak-coupling and strong-coupling cases. The transformation depends on avariationally determined parameter that characterizes the charge distribution across thepolaron volume. The relation between the spectral functions of the polaron and electron,the latter corresponding to the photoemission spectrum, is derived. Particular attention ispaid to the distinction between the coherent and incoherent parts of the spectra, and theirevolution as a function of band filling and model parameters. Results are discussedand compared with recent numerical calculations for the many-polaron problem.

Optical response of many-polaron systems

Exact results for the density of states and the AC conductivity of the spinless Holstein model at finite carrier density are obtained combining Lanczos and kernel polynomial methods.

Photoemission spectra of many-polaron systems

The crossover from low to high carrier densities in a many-polaron system is studied in the framework of the one-dimensional spinless Holstein model, using unbiased numerical methods. Combining a novel quantum Monte Carlo approach and exact diagonalization, accurate results for the single-particle spectrum and the electronic kinetic energy on fairly large systems are obtained. A detailed investigation of the quality of the Monte Carlo data is presented. In the physically most important adiabatic intermediate electron-phonon coupling regime, for which no analytical results are available, we observe a dissociation of polarons with increasing band filling, leading to normal metallic behavior, while for parameters favoring small polarons, no such density-driven changes occur. The present work points toward the inadequacy of single-polaron theories for a number of polaronic materials such as the manganites.

Polaronic and Nonadiabatic Phase Diagram from Anomalous Isotope Effects

Isotope effects (IEs) are powerful tools to probe directly the dependence of many physical properties on lattice dynamics. In this Letter we investigate the onset of anomalous IEs in the spinless Holstein model by employing the dynamical mean field theory. We show that the isotope coefficients of the electron effective mass and of the dressed phonon frequency are sizable also far away from the polaronic crossover and mark the importance of nonadiabatic lattice fluctuations. We draw a nonadiabatic phase diagram in which we identify a novel crossover, not related to polaronic features, where the IEs attain their largest anomalies.

Single-particle excitations and phonon softening in the one-dimensional spinless Holstein model

We investigate the influence of the electron-phonon coupling in the one-dimensional spinless Holstein model at half-filling using both a recently developed projector-based renormalization method (PRM) and an refined exact diagonalization technique in combination with the kernel polynomial method. At finite phonon frequencies the system shows a metal-insulator transition accompanied by the appearance of a Peierls distorted state at a finite critical electron-phonon coupling. We analyze the opening of a gap in terms of the (inverse) photoemission spectral functions which are evaluated in both approaches. Moreover, the PRM approach reveals the softening of a phonon at the Brillouin-zone boundary which can be understood as precursor effect of the gap formation.

Infrared conductivity of a one-dimensional charge-ordered state: Quantum lattice effects

The optical properties of the charge-ordering (CO) phase of the one-dimensional (1D) half-filled spinless Holstein model are derived at zero temperature within a well-known variational approach improved including second-order lattice fluctuations. Within the CO phase, the static lattice distortions give rise to the optical interband gap, that broadens as the strength of the electron-phonon (el-ph) interaction increases. The lattice fluctuation effects induce a long subgap tail in the infrared conductivity and a wide band above the gap energy. The first term is due to the multiphonon emission by the charge carriers and the second to the interband transitions accompanied by the multiphonon scattering. The results show a good agreement with experimental spectra of quasi-1D conductors such as ${\mathrm{K}}_{0.3}{\mathrm{MoO}}_{3}$ and $({\mathrm{TaSe}}_{4}{)}_{2}\mathrm{I}.$

Infrared absorption of the charge-ordering phase: Lattice effects

The phase diagram of the half-filled spinless Holstein model for electrons interacting with quantum phonons is derived in three dimensions extending at finite temperature T a variational approach introduced for the one-dimensional $T=0$ case. Employing the variational scheme, the spectral and optical properties of the system are evaluated in the different regimes that characterize the normal and ordered states. The effects of the charge ordering (CO) induce a transfer of spectral weight from low to high energies in the conductivity spectra, as the temperature decreases or the strength of the electron-phonon interaction increases. The inclusion of effects of lattice fluctuations is able to smooth the inverse square-root singularity expected for the case of the mean-field approach and determines a subgap tail absorption. Moreover, in the weak- to intermediate-coupling regime, a two-component structure is obtained within the CO phase at low frequency: the remnant Drude-like term and the incipient absorption band centered around the gap energy.

Nonlinear Lattice Effect on Ground State of One-Dimensional Electron-Phonon System**The project supported by the Research Foundation of Hunan Normal University under Grant No.200606

We study the nonlinear lattice effect on the ground state in a one-dimensional spinless Holstein model with nonadiabitical coupling and squeezing by means of the parameter variational approach. Our results show that the introduction of a hard quartic term in the lattice potential increases the ground state energy of the system when electron-phonon coupling is strong, and the increment is sensitive to the magnitude of the lattice quartic force constant. In this case the nonlinear lattice effects should be taken into account to describe satisfactorily some physical properties of the coupling system.

Phase diagram and optical conductivity of the one-dimensional spinless Holstein model

The effects of quantum lattice fluctuations on the Peierls transition and the optical conductivity in the one-dimensional Holstein model of spinless fermions have been studied by developing an analytical approach, based on the unitary transformation method. We show that when the electron-phonon coupling constant decreases to a finite critical value the Peierls dimerization is destroyed by the quantum lattice fluctuations. The dimerization gap is much more reduced by the quantum lattice fluctuations than the phonon order parameter. The calculated optical conductivity does not have the inverse-square-root singularity but have a peak above the gap edge and there exists a significant tail below the peak. The peak of optical-conductivity spectrum is not directly corresponding to the dimerized gap. Our results of the phase diagram and the spectral-weight function agree with those of the density matrix renormalization group and the exact diagonalization methods.

Gapped and gapless ground state of the one-dimensional spinless Holstein model

An analytical approach has been developed to study the ground-state phase diagram of a one-dimensional spinless Holstein model and its gapped and gapless phases. We show that when the electron-phonon coupling constant ${g}^{2}/{\ensuremath{\omega}}_{0}$ decreases or the phonon frequency ${\ensuremath{\omega}}_{0}$ increases the lattice dimerization and the gap in the fermionic spectrum decreases gradually. At some critical value ${(g}^{2}/{\ensuremath{\omega}}_{0}{)}_{c}$ or ${\ensuremath{\omega}}_{0c},$ the lattice dimerization disappears and the system becomes gapless Luttinger liquid with phonon-renormalized velocity $v(q)$ and stiffness constant $K(q).$ Our numerical results agree quite well with the results of Monte Carlo simulations.