Holstein Model Research Articles

Exact solution for the ground-state energy band of an extended one-dimensional Holstein model

Exact solution for the ground-state energy band of an extended one-dimensional Holstein model

Mpsqd: A matrix product state based Python package to simulate closed and open system quantum dynamics.

We introduce a Python package based on matrix product states (MPS) to simulate both the time-dependent Schrödinger equation (TDSE) and the hierarchical equations of motion (HEOM). The wave function in the TDSE or the reduced density operator/auxiliary density operators in the HEOM are represented using MPS. A matrix product operator (MPO) is then constructed to represent the Hamiltonian in the TDSE or the generalized Liouvillian in the HEOM. The fourth-order Runge-Kutta method and the time-dependent variational principle are used to propagate the MPS. Several examples, including the nonadiabatic interconversion dynamics of the pyrazine molecule, excitation energy transfer dynamics in molecular aggregates and photosynthetic light-harvesting complexes, the spin-boson model, a laser driven two-state model, the Holstein model, and charge transport in the Anderson impurity model, are presented to demonstrate the capability of the package.

Benchmarking the exponential ansatz for the Holstein model.

Polarons are quasiparticles formed as a result of lattice distortions induced by charge carriers. The single-electron Holstein model captures the fundamentals of single polaron physics. We examine the power of the exponential ansatz for the polaron ground-state wavefunction in its coupled cluster, canonical transformation, and (canonically transformed) perturbative variants across the parameter space of the Holstein model. Our benchmark serves to guide future developments of polaron wavefunctions beyond the single-electron Holstein model.

Tensor-Train Format Hierarchical Equations of Motion Formalism: Charge Transfer in Organic Semiconductors via Dissipative Holstein Models.

Hierarchical Equations of Motion (HEOM) in the Tensor-Train (TT) representation is applied to study the charge-transfer dynamics in organic semiconductors (OSCs). The theoretical formulation as well as the basic computational aspects of HEOM-TT are discussed in detail. Charge transfer in OSCs is modeled using dissipative polaronic models that incorporate the effects of both high- and low-frequency molecular vibrations, and it is simulated in a fully quantum and nonperturbative manner, which has not been studied intensively. The capability of treating complex electron-vibrational systems is examined by analyzing and comparing the numerical behavior of the time-dependent variational approach and the time-Alternating Minimal Energy methods and by calculating the current autocorrelation function and diffusivity across various models. Our results indicate that the HEOM-TT framework offers a robust tool for the detailed analysis of complex polaronic systems, suggesting its potential for broader applications.

Metal-insulator transition of spinless fermions coupled to dispersive optical bosons

Including the previously ignored dispersion of phonons we revisit the metal-insulator transition problem in one-dimensional electron-phonon systems on the basis of a modified spinless fermion Holstein model. Using matrix-product-state techniques we determine the global ground-state phase diagram in the thermodynamic limit for the half-filled band case, and show that in particular the curvature of the bare phonon band has a significant effect, not only on the transport properties characterized by the conductance and the Luttinger liquid parameter, but also on the phase space structure of the model as a whole. While a downward curved (convex) dispersion of the phonons only shifts the Tomonaga-Luttinger-liquid to charge-density-wave quantum phase transition towards stronger EP coupling, an upward curved (concave) phonon band leads to a new phase-separated state which, in the case of strong dispersion, can even completely cover the charge-density wave. Such phase separation does not occur in the related Edwards fermion-boson model.

Vertex corrections to conductivity in the Holstein model: A numerical-analytical study

Vertex corrections to conductivity in the Holstein model: A numerical-analytical study

Floquet nonadiabatic mixed quantum-classical dynamics in periodically driven solid systems.

In this paper, we introduce the Floquet mean-field dynamics and Floquet surface hopping approaches to study the nonadiabatic dynamics in periodically driven solid systems. We demonstrate that these two approaches can be formulated in both real and reciprocal spaces. Using the two approaches, we are able to simulate the interaction between electronic carriers and phonons under periodic drivings, such as strong light-matter interactions. Employing the Holstein and Peierls models, we show that strong light-matter interactions can effectively modulate the dynamics of electronic population and mobility. Notably, our study demonstrates the feasibility and effectiveness of modeling low-momentum carriers' interactions with phonons using a truncated reciprocal space basis, an approach impractical in real space frameworks. Moreover, we reveal that even with a significant truncation, carrier populations derived from surface hopping maintain greater accuracy compared to those obtained via mean-field dynamics. These results underscore the potential of our proposed methods in advancing the understanding of carrier-phonon interactions in various periodically driven materials.

Comparison of Different Thermostats in the Holstein Model

Comparison of Different Thermostats in the Holstein Model

Quench dynamics in higher-dimensional Holstein models: Insights from truncated Wigner approaches

Quench dynamics in higher-dimensional Holstein models: Insights from truncated Wigner approaches

Is there a finite mobility for the one vibrational mode Holstein model? Implications from real time simulations.

The question of whether there exists a finite mobility in the standard Holstein model with one vibrational mode on each site remains unclear. In this Communication, we approach this problem by employing the hierarchical equation of motion method to simulate model systems where the vibrational modes are dissipative. It is found that, as the friction becomes smaller, the charge carrier mobility increases significantly and a friction-free limit cannot be obtained. The current autocorrelation functions are also calculated for the friction-free Holstein model, and converged results cannot be obtained with an increase in the number of sites. Based on these observations, we conclude that a finite mobility cannot be defined for the standard Holstein model in the parameter regime explored in this work.

The impact of Rashba spin-orbit coupling in charge-ordered systems

We study the impact of the Rashba spin–orbit coupling (RSOC) on the stability of charge-density wave (CDW) in systems with large electron-phonon coupling (EPC). Here, the EPC is considered in the framework of the Holstein model at the half-filled square lattice. We obtain the phase diagram of the Rashba–Holstein model using the Hartree–Fock mean-field theory, and identify the boundaries of the CDW and Rashba metal phases. We notice that the RSOC disfavors the CDW phase, driving the system to a correlated Rashba metal. Also, we employ a cluster perturbation theory (CPT) approach to investigate the phase diagram beyond the Hartree–Fock approximation. The quantum correlations captured by CPT indicate that the RSOC is even more detrimental to CDW than previously anticipated. That is, the Rashba metal region is observed to be expanded in comparison to the mean-field case. Additionally, we investigate pairing correlations, and the results further strengthen the identification of critical points.

Anomalous Transport of Small Polarons Arises from Transient Lattice Relaxation or Immovable Boundaries.

Elucidating transport mechanisms is crucial for advancing material design, yet state-of-the-art theory is restricted to exact simulations of small lattices with severe finite-size effects or approximate ones that assume the nature of transport. We leverage algorithmic advances to tame finite-size effects and exactly simulate small polaron formation and transport in the Holstein model. We further analyze the applicability of the ubiquitously used equilibrium-based Green-Kubo relations and nonequilibrium methods to predict charge mobility. We find that these methods can converge to different values and track this disparity to finite-size dependence and the sensitivity of Green-Kubo relations to the system's topology. Contrary to standard perturbative calculations, our results demonstrate that small polarons exhibit anomalous transport that manifests transiently due to nonequilibrium lattice relaxation or permanently as a signature of immovable boundaries. These findings can offer new interpretations of transport experiments on polymers and transition metal oxides.

On the application of Marcus-Hush theory to small polaron chemical dynamics in oxides: its relationship to the Holstein model and the importance of lattice-orbital symmetries.

The chemical dynamics of small polaron hopping within oxides is often interpreted through two-site variations on Marcus-Hush theory, while from a physics perspective small polaron hopping is more often approached from Holstein's solid-state formalism. Here we seek to provide a chemically oriented viewpoint, focusing on small polaron hopping in oxides, concerning these two phenomenological frameworks by employing both tight-binding modelling and first-principles calculations. First, within a semiclassical approach the Marcus-Hush relations are overviewed as a two-site reduction of Holstein's model. Within the single-band regime, similarities and differences between Holstein derived small polaron hopping and the Marcus-Hush model are also discussed. In this context the emergence of adiabaticity (or, conversely, diabaticity) is also explored within each framework both analytically and by directly evolving the system wavefunction. Then, through first-principles calculations of select oxides we explore how coupled lattice and orbital symmetries can impact on hopping properties - in a manner that is quite distinct typical chemical applications of Marcus-Hush theory. These results are then related back to the Holstein model to explore the relative applicability of the two frameworks towards interpreting small polaron hopping properties, where it is emphasized that the Holstein model offers an increasingly more appealing physicochemical interpretation of hopping processes as band and/or coupling interactions increase. Overall, this work aims to strengthen the physically oriented exploration of small polarons and their physicochemical properties in the growing oxide chemistry community.

Particularities of polaron formation in the extended Holstein model with next nearest neighbor transfer

Particularities of polaron formation in the extended Holstein model with next nearest neighbor transfer

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.

Bipolaronic Nature of the Pseudogap in Quasi-One-Dimensional (TaSe4)2I Revealed via Weak Photoexcitation.

The origin of the pseudogap in many strongly correlated materials has been a longstanding puzzle. Here, we present experimental evidence that many-body interactions among small Holstein polarons, i.e., the formation of bipolarons, are primarily responsible for the pseudogap in (TaSe4)2I. After weak photoexcitation of the material, we observe the appearance of both dispersive (single-particle bare band) and flat bands (single-polaron sub-bands) in the gap by using time- and angle-resolved photoemission spectroscopy. Based on Monte Carlo simulations of the Holstein model, we propose that the melting of pseudogap and emergence of new bands originate from a bipolaron to single-polaron crossover. We also observe dramatically different relaxation times for the excited in-gap states in (TaSe4)2I (∼600 fs) compared with another 1D material Rb0.3MoO3 (∼60 fs), which provides a new method for distinguishing between pseudogaps induced by polaronic or Luttinger-liquid many-body interactions.

Holstein polaron transport from numerically "exact" real-time quantum dynamics simulations.

Numerically "exact" methods addressing the dynamics of coupled electron-phonon systems have been intensively developed. Nevertheless, the corresponding results for the electron mobility μdc are scarce, even for the one-dimensional (1d) Holstein model. Building on our recent progress on single-particle properties, here we develop the momentum-space hierarchical equations of motion (HEOM) method to evaluate real-time two-particle correlation functions of the 1d Holstein model at a finite temperature. We compute numerically "exact" dynamics of the current-current correlation function up to real times sufficiently long to capture the electron's diffusive motion and provide reliable results for μdc in a wide range of model parameters. In contrast to the smooth ballistic-to-diffusive crossover in the weak-coupling regime, we observe a temporally limited slow-down of the electron on intermediate time scales already in the intermediate-coupling regime, which translates to a finite-frequency peak in the optical response. Our momentum-space formulation lowers the numerical effort with respect to existing HEOM-method implementations, while we remove the numerical instabilities inherent to the undamped-mode HEOM by devising an appropriate hierarchy closing scheme. Still, our HEOM remains unstable at too low temperatures, for too strong electron-phonon coupling, and for too fast phonons.

Thermal and optical conductivity in the Holstein model at half filling and finite temperature in the Luttinger-liquid and charge-density-wave regime

Thermal and optical conductivity in the Holstein model at half filling and finite temperature in the Luttinger-liquid and charge-density-wave regime

Enhancement of charge density wave correlations in a Holstein model with an anharmonic phonon potential

Enhancement of charge density wave correlations in a Holstein model with an anharmonic phonon potential

Electronic Mechanism that Quenches Field-Driven Heating as Illustrated with the Static Holstein Model.

Time-dependent driving of quantum systems has emerged as a powerful tool to engineer exotic phases far from thermal equilibrium, but in the presence of many-body interactions it also leads to runaway heating, so that generic systems are believed to heat up until they reach a featureless infinite-temperature state. Understanding the mechanisms by which such a heat death can be slowed down or even avoided is a major goal-one such mechanism is to drive toward an even distribution of electrons in momentum space. Here we show how such a mechanism avoids runaway heating for an interacting charge-density-wave chain with a macroscopic number of conserved quantities when driven by a strong dc electric field; minibands with nontrivial distribution functions develop as the current is prematurely driven to zero. Moreover, when approaching a zero-temperature resonance, the field strength can tune between positive, negative, or close-to-infinite effective temperatures for each miniband. Our results suggest that nontrivial metastable distribution functions should be realized in the prethermal regime of quantum systems coupled to slow bosonic modes.