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Polaron Crossover and Bipolaronic Metal-Insulator Transition in the half- filled Holstein model

The formation of a finite density multipolaronic state is analyzed in the context of the Holstein model using the Dynamical Mean-Field Theory. The spinless and spinful fermion cases are compared to disentangle the polaron crossover from the bipolaron formation. The exact solution of Dynamical Mean-Field Theory is compared with weak-coupling perturbation theory, non-crossing (Migdal), and vertex correction approximations. We show that polaron formation is not associated to a metal-insulator transition, which is instead due to bipolaron formation.

More specifically, polaronic features have been observed in lightly doped cuprates [5], in the manganites [2], up to some indication in the fullerenes [6]. A small polaron is a carrier so tightly interacting with the lattice that its effective mass is strongly enhanced therefore reducing its mobility [7]. The single polaron problem has been extensively studied allowing to understand in detail the polaron physics [8,9]. Nevertheless, the experimental findings of polaronic effects obviously deal with finite densities of carriers, and prompt for an analogous understanding of the finite density polaron problem, where the polarons are able to interact and to change drastically the phonon properties. In this regard, it is important to stress the distinction between polaronic and bipolaronic states. If repulsion between the electrons is neglected, two polarons (with opposite spin) tend in fact to bind, giving rise to a (local) pair, which is called bipolaron. The bipolarons in turn may undergo a superconducting transition of the Bose-Einstein type [10]. If superconductivity is not allowed, bipolarons give rise to an insulating state of localized pairs [11,12] which may eventually condense in a charge-ordered state [13]. In some important compounds bipolaronic states are indeed unlikely formed. In cuprates and fullerenes, the strong electron correlation forbids bipolaron formation, while in the manganites, the double-exchange mechanism favors ferromagnetic states at low temperature, in which no bipolaron can be formed.
In order to disentangle the polaron effects from bipolaron formation, we compare the spinless fermion case, in which bipolarons are forbidden by Pauli principle, with the spinful case, which has been extensively studied in the recent past [14,15,16,17,18]. To be more explicit, we show that the polaron crossover at finite density is not by itself a metal insulator transition (MIT), and insulating behavior can only be associated to localized bipolarons. This effect shows up in the non-vanishing of both the quasiparticle renormalization factor, and of the renormalized phonon frequency in the spinless case. In the spinful case the quasiparticle weight vanishes at some definite coupling while the renormalized phonon frequency remains finite.
We consider the Holstein molecular crystal model, in which tight-binding electrons interact with local modes of constant frequency. The Hamiltonian is where c i,σ (c † i,σ ) and a i (a † i ) are destruction (creation) operators for fermions and for phonons of frequency ω 0 , n i the electron density, t is the hopping amplitude, g is an e-ph coupling. The density is fixed to n = 0.5 (n = 1) in the spinless (spinful) system, which corresponds to the particle-hole symmetric half-filled case. In analogy with the studies of the Mott transition in the Hubbard model, in which antiferromagnetism is neglected [19], we restrict ourselves to the state with no charge order [12]. Despite in the particle-hole symmetric case the ground state is always ordered, the homogeneous solution may become representative of the groundstate whenever charge ordering is spoiled by some frustration effect like a nextnearest-neighbor hopping. Moreover, this study allows to characterize how strong e-ph interaction may lead to the destruction of the metallic state (just like electron correlation leads to the Mott state). The adiabatic ratio γ = ω 0 /t has been shown to be an important parameter for the single polaron formation. In the adiabatic regime γ < 1, the polaron crossover occurs when λ = g 2 /ω 0 t ≃ 1, while in the antiadiabatic regime γ > 1, the condition is instead (g/ω 0 ) 2 ≃ 1 [8,9]. We solve the model using the Dynamical Mean-Field Theory (DMFT), a non-perturbative approach which becomes exact in the limit of infinite dimensions [19]. We notice that in such a limit the spinless fermion case does not coincide with the infinite correlation limit. Nonetheless, this system represents an instructive playground where polaronic effects can be observed without too many competing phases. In DMFT, the lattice model is mapped onto an impurity problem subject to a selfconsistency condition, which contains the information about the lattice. In our case the impurity model is where the phonons live only on the impurity (f ) site, E k and V k are the energy levels and the hybridization parameters of the conduction bath. For the z-coordination Bethe lattice of half-bandwidth D = 2t √ z = 1 the selfconsistency equation in the z → ∞ limit is In this work we use Exact Diagonalization (ED) to solve the impurity model (2) [20]. This method requires to restrict the sum in Eqs. (2,3) to a finite small number of levels N s − 1. The discretized model can then be solved at T = 0 using the Lanczos method. The convergence is exponential in N s , and a few levels are enough to give converged results. Results are obtained taking typically N s = 10, having checked that no significant change occurs for larger N s [21]. Error bars can be evaluated by linearly extrapolating (in 1/N s ) to N s → ∞, and are of the size of symbols in the figures. We compare the ED solution of DMFT, with two approximate schemes: a selfconsistent Non Crossing Approximation (NCA) [11,15] (diagrams (a) in Fig. 1) and a self-consistent approximation including the first Vertex Correction beyond NCA (VCA) [15] (diagrams (a) and (b) in Fig. 1). Notice that in the standard Migdal-Eliashberg approximation the phonon spectrum is not self-consistently evaluated, but it is taken "from experiments", while in our NCA and VCA the phonon self-energy is self-consistently evaluated (on the imaginary frequency axis) through an iterative scheme, just like the electron self-energy. Actually, to allow for a safer truncation on the frequency axis, the quantity which is iteratively determined is not the full self-energy, but the difference between this quantity and the result from second order perturbation theory. Zero temperature results are obtained by lowering the temperature until the physically relevant quantities converge (consistently increasing the number of Matsubara frequecies N max = 6/(2T π)). We checked the convergence to T = 0 by monitoring the quasiparticle spectral weight Z defined as where ω n = (2n − 1)πT . The vanishing of Z is used within DMFT to characterize the Mott MIT in the repulsive [19], and the pairing transition in the attractive Hubbard models [22]. A vanishing Z has been found also in the spinful Holstein model [12]. The convergence to T = 0 turns out to depend on both γ (as detailed below, we study γ = 0.1 and γ = 1 as representative of adiabatic and nonadiabatic regimes, respectively) and λ. Within VCA T /t = 3 × 10 −3 is sufficient to get results representative of the ground state for weak/intermediate coupling. In the spinful case for γ = 0.1 T = 10 −3 is instead necessary since the polaron crossover is approached for smaller coupling (see below). Within NCA it is possible to span the strong coupling regime, making an extrapolation to T = 0 necessary in the adiabatic regime for λ > 1 (λ > 0.5) in the spinless (spinful) case. In the nonadiabatic regime the extrapolation is required for λ > 1.6 (λ > 1.2) in the spinless (spinful) case. We first discuss the spinless case. Exact DMFT results for Z are shown in Fig. 2. The logarithmic scale on the yaxis evidences that in both cases Z, even if exponentially reduced, does never vanish by increasing λ, indicating that no MIT is taking place. It is important to observe that Z increases when the truncation error is reduced increasing N s . The comparison with NCA and VCA shows non trivial tendencies. Both approximations are accurate at weak coupling, but VCA remains closer to ED for relatively large coupling even in the adiabatic regime, where the Migdal approximation would be expected to hold. Quite surprisingly even in the non-adiabatic regime VCA improves NCA only at weak coupling. At strong coupling NCA predicts a polaronic crossover, in qualitative agreement with exact results, even if, for γ = 1, the crossover coupling is strongly underestimated by NCA. On the other hand, VCA drastically diverges from ED, and gives unphysical negative mass renormalization at strong coupling both for γ = 0.1 and 1, in agreement with the divergence of vertex corrections predicted in Ref. [11]. The coupling at which the approximate methods deviate from exact results decreases with increasing γ, and it is unrelated with the polaron crossover coupling which is instead larger for γ = 1, as shown in the inset of Fig.  1, and in agreement with the single polaron case [8,9].
An important difference between the finite-density situation and the single polaron problem is that many electrons are able to renormalize the phonon properties. The renormalized phonon frequency can be obtained from the phonon propagator D(iω n ) = β 0 dτ e iωnτ T (a 0 (τ ) + a † 0 (τ ))(a 0 + a † 0 ) , as (Ω/ω 0 ) 2 = −2/ω 0 D(iω n = 0) −1 . As it is shown in Fig. 3, Ω/ω 0 never vanishes as a function of the coupling, but rather exponentially decreases. The way NCA and VCA results for Ω compare with ED is analogous to the results for Z. In the adiabatic regime NCA strongly overestimates Ω/ω 0 around the polaron crossover. In the nonadiabatic regime NCA again predicts a phonon softening which does not actually occur at such small values of the coupling. In the same regime VCA improves NCA result only at weak coupling and gives a phonon hardening at strong coupling. It is worth to note that for both Z and Ω the coupling at which NCA and VCA deviate from ED is lower in nonadiabatic than in adiabatic regime. This "non uniform" behavior is not present for a single polaron [9], and represents a first peculiarity of the finite-density case. Now we compare our findings for spinless fermions with the spinful fermion case. In the adiabatic regime a MIT has been found around λ ≃ λ MIT ≃ 0.68 [17], close to the value (λ = 0.664) at which the density of state at Fermi energy vanishes in the adiabatic limit (γ = 0) [16]. This transition has a precursor in the phonon softening [12] but, contrary to the spinless case, the spin degrees of freedom prevent electron coherent hopping at strong coupling through a Kondo-like mechanism [23]. A MIT has been claimed to occur also within NCA [11].
In Fig. 4 we show results for γ = 0.1. The logarithmic plot clearly shows that Z vanishes faster than exponential, signaling a MIT at λ ≃ 0.76, in agreement with Ref. [12]. Contrary to the spinless case, here Z decreases by increasing N s . On the other hand, Ω/ω 0 decreases with λ as in the spinless case, suggesting that, even if strongly softened, the phonon mode never becomes completely soft. This behavior is more evident for larger phonon frequency γ = 1, as reported in Fig. 5. Also in this case Z vanishes much faster than exponential at λ = 1.44, where the MIT takes place, while the phonon renormalization is much less effective and leads to quite a large value of Ω/ω 0 even at the MIT point (see inset).
The comparison with approximate schemes qualitatively resembles the spinless case. NCA underestimates Z for λ below the MIT, but it gives a finite weight also above the exact MIT point. Our extrapolation to T = 0 from finite temperatures does not allow us to definitively rule out the existence of a MIT within NCA even if, in contrast with a claim in Ref. [11], Z seems to decrease exponentially with λ within this approximation [24]. Again VCA gives better results than NCA even in the adiabatic regime up to intermediate coupling λ ≃ 0.55, but it becomes completely unreliable at strong coupling.
We have studied the formation of a polaronic state in the Holstein model at half-filling within DMFT. For spinless fermions, a continuous crossover leads to a polaronic state by increasing the coupling constant. Despite the electron effective mass becomes exponentially large in the strong-coupling regime, the ground state is always metallic. The crossover is more abrupt in the adiabatic case. In the spinful case, the polarons can bind to form bipolarons, leading to a real MIT. The phonon renormalization is much stronger in the adiabatic regime than in the nonadiabatic case, but the phonons do not become completely soft at the MIT. Approximate treatments (NCA and VCA) strongly deviate from exact DMFT above a coupling which diminishes with increasing ω 0 /t. At weak coupling VCA correctly reproduces the qualitative trends of exact results, and improves on NCA.
M. C. thanks the hospitality and financial support of the Physics Department of the University of Rome "La Sapienza", and of INFM, Unitá Roma 1. We acknowledge financial support of MIUR Cofin 2001 and enlightening discussions with C. Castellani.