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Mott transition in the Hubbard model away from particle-hole symmetry

We solve the Dynamical Mean Field Theory equations for the Hubbard model away from the particle-hole symmetric case using the Density Matrix Renormalization Group method. We focus our study on the region of strong interactions and finite doping where two solutions coexist. We obtain precise predictions for the boundaries of the coexistence region. In addition, we demonstrate the capabilities of this precise method by obtaining the frequency dependent optical conductivity spectra.

Introduction: Solving models for strongly correlated compounds (typically those including d or f electrons) remains a hard and important open problem of modern condensed matter physics. In these systems interesting anomalous behavior usually occurs as a consequence of the competition between the kinetic and Coulomb energy of electrons, which are of the same order of magnitude, and analytical methods based on perturbative considerations are notoriously unreliable. Therefore, one is led to resort to non-perturbative techniques and numerical methods to try to deal with these difficulties.
In recent years, a non-perturbative approach to general strongly correlated electron models, termed Dynamical Mean Field Theory (DMFT), 1,2 allowed progress in the understanding of several physical problems, including the correlation-driven Mott metal-insulator transition. The key feature of DMFT is that it maps the original lattice problem onto a selfconsistent quantum impurity model. This resulting quantum impurity remains, nevertheless, a fully interacting many-body problem that has to be solved. 1 The success of DMFT in dealing with model Hamiltonians has generated a great deal of interest in combining it with ab initio band structure methods with the goal of obtaining a realistic description of correlated electron compounds. 2 The main technical difficulty is the lack of a reliable method to solve the associated quantum impurity problem. Currently, the most widely adopted methodology is the quantum Monte Carlo technique, which generally requires an analytical continuation of the results to the real frequency axis. Unfortunately, the latter introduces some uncertainty in the procedure, which becomes a severe problem for multi-orbital systems. Therefore, there is strong interest in the development of new methods that can deal with general quantum impurity models directly on the real axis. An interesting proposal was recently introduced which was based on the precise diagonalization of the quantum impurity Hamiltonian with the powerful Density Matrix Renormalization Group (DMRG). 3,4,5 This technique has the virtue of being already widely employed in the study of low-dimensional strongly correlated systems and is based on a judicious trimming of the Hilbert space so as to restrict the numerical calculation to its most relevant subspace. 6 In this manner, despite the exponential growth of the Hilbert space, hundreds of orbitals can be taken into account.
Moreover, the method is not restricted to the ground state and finite energy excitations can be computed with similar accuracy. 7 This is in contrast to the related alternative, the Numerical Renormalization Group (NRG) method, which aims at a very precise description of the low frequency quasiparticle peaks associated with low-energy excitations, as done in its celebrated application to the Kondo problem. 8 The DMRG, on the other hand, treats high and low frequencies on an equal footing.
The initial DMRG solution of the DMFT equations was obtained for the simplest test case of the particle-hole symmetric Hubbard model. 3,4,5 In this paper we show that the method can reliably tackle the general finite doping case. In particular we focus on the most demanding region of the phase diagram where two solutions coexist near the correlation-driven Mott metal-insulator transition and obtain the phase boundaries with unprecedented precision.
We also illustrate the capabilities of the methodology by computing the frequency-dependent optical conductivity, which requires the reliable description of higher energy features, such as the Hubbard bands, that lie beyond the scope of the NRG method. Our results show that even at the smallest dopings and strong interaction strength, the low frequency contribution to the optical conductivity, the Drude part, remains very well described by a Lorentzian line-shape. We also show that both the vanishing of the Drude weight or the doping can signal equally well the destruction of the correlated Fermi liquid metallic state. Our work amounts to a substantial improvement with respect to previous studies based on exact diagonalization (of up to 8 sites, however nowadays somewhat larger systems are possible with ED) and an ad hoc perturbation scheme, 9 and opens the way for the application of the DMFT+DMRG method to more general model Hamiltonians. Details of the method will be given elsewhere 10 .
The Model:The Hamiltonian of the Hubbard model is defined by where U is the on-site Coulomb interaction, µ is the chemical potential, t is the hopping and d is the space dimension. We take the half bandwidth of the non interacting model as unit of energy, thus D = 2t = 1. We particularize to the case of the infinite-dimensional Bethe lattice, in which the non-interacting density of states (DOS) is The associated impurity problem is the single-impurity Anderson model (SIAM) and its hybridization function needs to be self-consistently determined. 1 The SIAM Hamiltonian is solved in linear chains 11,12 using the DMRG algorithm 3 of up to 101 sites keeping 128 states per block.
We focus our study on the poorly explored region of the U -µ parameter space where two solutions, insulating and metallic, can be obtained from the DMFT equations. In Fig. 1 we show the evolution of the DOS for the two solutions as one moves away from the half-filled particle-hole symmetric case. The chemical potential µ is increased at fixed U. While the use of DMRG allows the precise diagonalization of the associated SIAM problem with a bath that can be accurately described using around one hundred sites, this impurity model remains, nevertheless, finite. Therefore, the computed Green's functions contain a finite, though large, number of poles. Thus, as in any exact diagonalization scheme, one needs to broaden the poles to allow the observation of the DOS structure on the real axis.
Results: We show in Fig. 1 the results obtained by using a simple Lorentzian broadening η. We find that choice to be more appropriate than the "logarithmic" broadening usually adopted in NRG calculations 13,14 which, although capable of sharply resolving the insulating gap, tends to wash out the high energy features of the density of states. The results show that, in the insulating case, when the chemical potential is moved within the Mott gap, the lower and upper Hubbard bands shift rigidly, without any ostensible transfer of spectral weight taking place ( Fig. 1(a)). The apparent substructure in the Hubbard bands seen in the insulating DOS results from finite-size effects 15 , i.e., a finite number of poles. Our finite size analysis (not shown) suggests that in the infinite chain limit the Hubbard bands become smooth. This is in contrast to what is seen in the large U/D antiferromagnetic phase. 16 In the lightly doped case one observes that, as the central quasiparticle peak rapidly moves through the region between the Hubbard bands, there is a transfer of spectral weight as well as an evolution of the line shapes ( Fig. 1(b)). More precisely, one finds that the quasiparticle peak receives spectral weight from both Hubbard bands. For larger values of µ, as the system gets heavily doped, one finds that the quasiparticle peak eventually broadens as it merges with the closest Hubbard band (Fig. 1(c)). As these features coalesce, they also draw spectral weight from the other Hubbard band that remains at an energy distance of the order of U. 1 In order to demonstrate the capabilities of the method we shall now compute the frequency-dependent optical conductivity. From the lattice Green's G(ǫ k , ν) function, where ǫ k is the non-interaction dispersion, we can evaluate the optical conductivity within DMFT as 1,17 where a is the lattice spacing, d is the spatial dimension, ρ(ǫ, ν) = ImG(ǫ, ν − I0 + )/π, where τ is the relaxation time and DW is the Drude weight, which is a measure of the number of quasiparticle carriers in the metal. 19 In our metallic case the finite value of τ −1 comes from the finite imaginary part used to compute the Green's function G. The inset shows that as η goes to zero, τ −1 tends to zero and we recover the delta-function behavior that corresponds to a clean Fermi Liquid. For large U/D we observe that, in addition to the small Drude part, the optical conductivity spectrum has a large mid-infrared contribution at frequencies of order U. This regular part corresponds to finite frequency optical excitations between the two Hubbard bands and between the latter and the central quasiparticle peak and is almost absent for small values of the Coulomb interaction.
The destruction of a normal metallic state is formally defined by the vanishing of the Drude weight, which signals the localization of all metallic conduction carriers. 20 On the other hand, in the Hubbard model it is well known that, unless in phases with some type of long range order or in the presence of disorder, infinitesimal doping at T = 0 is enough to drive the system across a density driven metal-insulator transition. Therefore, a demanding test for the present method would be to verify that both the doping δ = (n − 1)/2 (where   Fig. 3(b). Its behavior is qualitatively different from that of δ, since DW (U) does not uniformly increase with increasing µ. In the low U/D region, the DW decreases as the chemical potential is increased, reflecting the lowering of the kinetic energy due to the fewer number of carriers. In contrast, for larger values of the interaction close to the µ c2 line, the DW decreases as µ decreases towards particle-hole symmetry, reflecting the enhancement of the effective mass as the metal-insulator transition is approached.
It is also possible to investigate the instability of the insulating state towards the metal. The critical lines allow us to accurately draw the phase diagram of the model away from particle-hole symmetry at T = 0. Previous work has been restricted to either a small number of sites 9 in the effective impurity Hamiltonian or to small but finite temperatures using quantum Monte Carlo. 21 On the other hand, the NRG method is not very well suited for the accurate investigation of the insulating state. In Fig. 4 we plot the µ c1 (U) and µ c2 (U) lines that determine three regions in the µ-U phase diagram: For µ > µ c1 (U) (µ < µ c2 (U)) only metallic (insulating) solutions are found. In the middle there is a region of coexistence of both kinds of states, where the metallic state is one with the lowest energy. 1 The phase diagram presented here shows an overall agreement with the one obtained through exact diagonalization in the "star geometry" 9 where the impurity site is connected with hopping terms to all the other sites. The main differences are found for the µ c2 (U) line because, as the metal to insulator transition is approached, the quasiparticles develop a diverging mass corresponding to a very narrow quasiparticle peak. In the language of the associated SIAM, this narrow resonance implies a large correlation length which can be fully realized only in long enough systems. This can only be obtained with the method presented here, allowing for very accurate results.
Conclusions: In this work we have shown that the DMRG method, in addition to being largely used to compute spectral quantities of low dimensional strongly correlated systems, 6