Two-site Holstein Model Research Articles

Variations on the “exact factorization” theme

In a series of publications, Hardy Gross and co-workers have highlighted the interest of an “exact factorization” approach to the interacting electron-nuclei problem, be it time-independent or time-dependent. In this approach, an effective potential governs the dynamics of the nuclei such that the resulting N-body nuclear density is in principle exact. This contrasts with the more usual adiabatic approach, where the effective potential leads to an approximate nuclear density. Inspired by discussions with Hardy, we explore the factorization idea for arbitrary many-body Hamiltonians, generalizing the electron-nuclei case, with a focus on the static case. While the exact equations do not lead to any practical advantage, they are illuminating, and may therefore constitute a suitable starting point for approximations. In particular, we find that unitary transformations that diagonalize the coupling term for one of the sub-systems make exact factorization appealing. The algorithms by which the equations for the separate subsystems can be solved in the time-independent case are also explored. We illustrate our discussions using the two-site Holstein model and the quantum Rabi model. Two factorization schemes are possible: one where the boson field feels a potential determined by the electrons, and the reverse exact factorization, where the electrons feel a potential determined by the bosons; both are explored in this work. A comparison with a self-energy approach is also presented.

The Holstein polaron problem revisited

The Holstein Hamiltonian was proposed half a century ago; since then, decades of research have come up empty handed in the pursuit of a closed-form solution. An exact solution to the two-site Holstein model is presented in this paper. The obtained results provide a clear image of the Hamiltonian structure and allow for the investigation of the symmetry, energy level crossings and polaronic characteristics of the system. The main mathematical tool is a three-term recurrence relation between the wave function amplitudes, which was obtained using the properties of a family of orthogonal functions, namely the Poisson–Charlier polynomials. It is shown that, with the appropriate choice of basis, the eigenfunctions of the problem naturally fall into two families (parities) associated with the discrete symmetry of the Hamiltonian. The asymptotic solution to the recurrence relation is found by using the Birkhoff expansion. The asymptotic sets the truncation criterion for the wave function, which ensures the accurate calculation of the energy levels for any strength of electron–phonon interaction. The level crossing of states with different parities is discussed and the exact points of broken symmetry are found analytically. The results are used as the building blocks for studying a four-site system. The inherited symmetries lead to the formation of a sparse matrix that is convenient for numerical calculations.

Quantum entanglement and fidelity of the two-site Holstein model

We study the polaronic crossover properties in the two-site Holstein model by the quantum entanglement and the fidelity. Based on the exact wave function obtained by the extended bosonic coherent states, the linear entropy and the fidelity are evaluated. The maximum of the entanglement between the electron and surrounding phonons is observed in a small polaron regime. A smooth change (no singularity) of the ground-state fidelity as well as its susceptibility appears in the intermediate coupling regime, which confirms a crossover rather than a phase transition in this system by this quantum information tool. In addition, the results for some quantities in the weak-coupling and strong-coupling limit are given analytically.

A density matrix approach to the dynamical properties of a two-site Holstein model

The two-site Holstein model represents a first non-trivial paradigm for the interaction between an itinerant charge with a quantum oscillator, a very common topic in different ambits. Exact results can be achieved both analytically and numerically, nevertheless it can be useful to compare them with approximate, semi-classical techniques in order to highlight the role of quantum effects. In this paper we consider the adiabatic limit in which the oscillator is slower than the electron. A density matrix approach is introduced for studying the charge dynamics and the exact results are compared with two different approximations: a Born-Oppenheimer-based Static Approximation for the oscillator (SA) and a Quantum-classical (QC) dynamics.

Exact solutions of the Holstein model with different site energies

The two-site Holstein model with different site energies, the simplest model to mimic disorder, is analytically studied by a diagonalization method of coherent states. The solutions obtained in our work are exact, and agree well with those derived by a modified Lang-Firsov (MLF) method in the weak- and strong-coupling regimes. The deviation from the MLF solution in the intermediate-coupling regime implies that our solution, including the higher order correlation terms overlooked in the MLF treatment, gives a more accurate description. To check the validity of our approach, we have also calculated the case with the same site energies, which shows that our exact solution could be consistent with a previous treatment of the Holstein model using the coherent expansion method [Phys. Rev. B 65, 174303 (2002)]. The method and the results in the present work would be useful for testing various approximate methods for the Holstein model, and are applicable to more complicated situations.

Two-Site Holstein Model: a Variational Wavefunction Analysis

A novel variational approach is proposed to calculate the ground-state(GS) properties of the two-site Holstein model. By the linearsuperposition of two coherent states, which simulate the behaviour of theweak and strong coupling limits, we can obtain very accurate GS energyfor arbitrary electron–phonon coupling constant. Other GS properties arealso discussed. Moreover, the present concise approach is hopefullygeneralized to many other Holstein models.

A New Variational Study for Two-Site Holstein Model

A variational approach is developed to study the groundstate (GS) of thetwo-site Holstein model. By the extended coherent state, where the morephonon correlations are easily incorporated, we can get the very accurateground state energy for all electron-phonon coupling range in typical valuesof hopping integral t = 0.5,1.1, and 2.1 (in units of phonon frequencyω0), which covers the crossover region fromantiadiabatic limit to the adiabatic limit. Within a very wide trange [0,2.7], the exact results for the GS energy are obtainedwith the twelfth (fourteenth) order corrections to the zeroth order wavefunction. Moreover, the present approach is more concise than anyother analytical ones in this field, and hopefully can be easilygeneralized to many other Holstein models.

Analytical Solutions of the Two-Site Holstein Model by the Coherent State Expansion

The polaron problem presents a highly nonlinear dynamical system in which the charge and lattice deformations are intricately coupled together. The coherent state expansion method was applied to the two-site Holstein model. Analytical solutions are found for the first, the second and the third order approximations. Comparison is made among the exact numerical results, based upon solving a recursive relation for the expansion coefficients, and the analytical solutions up to the third order approximation for various e-ph coupling strengths and electron hopping parameters.

Exact solutions for the two-site Holstein model

The two-site Holstein model is studied by the analytical method of the coherent-state expansion. We obtain the recursive relationship among the expansion coefficients, which leads to the exact solutions of the eigenstate properties. The exact ground-state results in the whole coupling strength region and from antiadiabatic to adiabatic limits are in perfect agreement with that available by the exact numerical diagonalization technique. Comparison with the best perturbation results shows the success and shortcoming of the perturbative method. The coherent-state expansion method is a promising approach in studying electron-phonon $(e\ensuremath{-}\mathrm{ph})$ coupling systems.

FIRST EXCITED STATE CALCULATION USING DIFFERENT PHONON BASES FOR THE TWO-SITE HOLSTEIN MODEL

The single-electron energy and static charge-lattice deformation correlations have been calculated for the first excited state of a two-site Holstein model within perturbative expansions using different standard phonon bases obtained through Lang–Firsov (LF) transformation, LF with squeezed phonon states, modified LF, modified LF transformation with squeezed phonon states, and also within weak-coupling perturbation approach. Comparisons of the convergence of the perturbative expansions for different phonon bases reveal that modified LF approach works much better than other approaches for major range of the coupling strength.

Polaron dynamics and bipolaron condensation in cuprates

Based on the exact cluster diagonalization and recent Quantum Monte Carlo simulations we analyze dynamic properties of small polarons and bipolarons formed by short-range (Holstein) and long-range (Fr\ohlich) electron-phonon interactions. We show that the exact results agree well with canonical Holstein theory for a cluster and with Lang-Firsov theory for a lattice. Lang-Firsov theory of a single polaron and our $1/\ensuremath{\lambda}$ perturbation expansion for a multipolaron system are practically exact in a wide range of the adiabatic parameter $\ensuremath{\omega}/t$ and the electron-phonon coupling \ensuremath{\lambda} for a long-range interaction. (Bi)polarons exist in the itinerant Bloch states at temperatures below the characteristic phonon frequency no matter which values the parameters of the system take. We show that recent claims by several authors with regards to a breakdown of Holstein-Lang-Firsov theory of a small polaron and the ``impossibility'' of bipolaronic superconductivity are the result of an erroneous interpretation of the electronic energy levels of the two-site Holstein model and a misunderstanding of the electron-phonon interaction in ionic solids with polaronic carriers. A ``phase'' diagram in $t/\ensuremath{\omega}\ensuremath{-}\ensuremath{\lambda}$ space is proposed to elucidate the BCS and (bi)polaronic domains. Bipolaron theory provides a parameter-free expression for the superconducting critical temperature of layered cuprates. Crystallization of the (bi)polaronic liquid is shown to be impossible in the range of the parameters typical for cuprates. The small Fr\ohlich polaron has spectral features compatible with single-particle tunneling and photoemission in cuprates.

Comparison of perturbative expansions using different phonon bases for the two-site Holstein model

The two-site single-polaron problem is studied within perturbative expansions using different standard phonon bases obtained through Lang-Firsov (LF), modified LF, and modified LF transformations with squeezed phonon states. The role of these convergent expansions using the above prescriptions in lowering the energy and in determining the correlation functions are compared for different values of the coupling strength. The single-electron energy, oscillator wave functions, and correlation functions are calculated for the same system. The applicability of different phonon bases in different regimes of the coupling strength, as well as in different regimes of hopping, is also discussed.

Comment on “Dynamical properties of small polarons”

We show that the conclusion on the breakdown of the standard small polaron theory made recently by E.V. deMello and J. Ranninger (Phys. Rev. B 55, 14872 (1997)) is a result of an incorrect interpretation of the electronic and vibronic energy levels of the two-site Holstein model. The small polaron theory, when properly applied, agrees well with the numerical results of these authors. Also we show that their attempt to connect the properties of the calculated correlation functions with the features of the intersite electron hopping is unsuccessful.

Spectral properties of the small polaron

We compute exactly both the spectral function of the electron and of the small polaron for the two-site Holstein model in the case where phonon and electron energies are of the same order. We find that for intermediary coupling, the small polaron is a better fundamental excitation of the system than the electron. However, the Lang-Firsov approximation fails to predict accurately the dispersion relation for the small polaron.

Electronic and phononic transitions in the two-site Holstein model

We consider the Hamiltonian for a dimer including all the electronic one- and two-body terms for a single orbital, the free phonon term for a single frequency Ω, and an electron-phonon coupling g of the Holstein type. The bare electronic interaction parameters were evaluated in terms of Wannier functions, built from atomic Gaussian orbitals. An effective polaronic Hamiltonian was obtained by a displaced-oscillator transformation, followed by evaluation of the phononic terms over a squeezed-phonon variational wavefunction. Its eigenvalues and eigenvectors have been obtained in explicit form forN = 1, 2, 3, 4 electrons. For eachN and over a range of dimer length values, the ground state for given g and Ω was identified by optimizing the orbital shape and the squeezing effect strength. We find, for anyN ifg ≠ 0, an abrupt transition from a delocalized to a localized polaron phase, due to a discontinuous collapse of the squeezing effect above a critical dimer length. Also, in the caseN = 2, and for a different dimer length, we find a second transition from a singlet to a triplet ground state that depends also ong.All the effective electronic interactions are appreciably renormalized across the transitions, as the equilibrium shape of the wavefunctions strongly depends on both the squeezing effect and the type of ground state.

Coupled quasiparticle-boson systems: The semiclassical approximation and discrete nonlinear Schrödinger equation

The validity of the semiclassical approximation is studied for a system comprising one quasiparticle coupled to a boson degree of freedom. Using a two-site Holstein model as an example, it is shown that the semiclassical approximation becomes exact in a nontrivial adiabatic limit. Furthermore, in the model`s polaron regime, there exists a hierarchy of time scales that rationalizes the quantum dynamics of the Holstein model. For the single-mode case considered, the discrete nonlinear Schroedinger equation is found to be valid only in a highly limited antiadiabatic regime.