Phase competition in a one-dimensional three-orbital Hubbard-Holstein model

Phase competition in a one-dimensional three-orbital Hubbard-Holstein model

Shaozhi Li Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA    Yanfei Tang Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA    Thomas A. Maier Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA    Steven Johnston Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA Joint Institute of Advanced Materials at The University of Tennessee, Knoxville, Tennessee 37996, USA
July 15, 2019

We study the interplay between the electron-phonon (e-ph) and on-site electron-electron (e-e) interactions in a three-orbital Hubbard-Holstein model on an extended one-dimensional lattice using determinant quantum Monte Carlo. For weak e-e and e-ph interactions, we observe a competition between an orbital-selective Mott phase (OSMP) and a (multicomponent) charge-density-wave (CDW) insulating phase, with an intermediate metallic phase located between them. For large e-e and e-ph couplings, the OSMP and CDW phases persist, while the metallic phase develops short-range orbital correlations and becomes insulating when both the e-e and e-ph interactions are large but comparable. Many of our conclusions are in line with those drawn from a prior dynamical mean field theory study of the two-orbital Hubbard-Holstein model [Phys. Rev. B 95, 12112(R) (2017)] in infinite dimension, suggesting that the competition between the e-ph and e-e interactions in multiorbital Hubbard-Holstein models leads to rich physics, regardless of the dimension of the system.

71.30.+h, 71.10.Fd, 71.27.+a

I Introduction

The multiorbital iron-based superconductors (FeSCs) remain as a significant research problem in the condensed matter physics [JohnstonAP, ; Stewart, ; DaiNat, ]. While most of the FeSCs host quasi-two-dimensional FeAs or FeSe layers, a quasi-one-dimensional (1D) structure has been discovered recently in [Takahashinature, ], which has spurred interest in quasi-1D multiorbital models [NiravPRB2016, ; KaushalPRB, ; RinconPRL, ; LiuPRE, ]. has the geometry of a two-leg ladder, and its low-temperature magnetic structure is ferromagnetic (FM) in the ladder direction and antiferromagnetic (AFM) in the leg direction [Takahashinature, ; NiravPRB2016, ]. Based on its magnetic properties, and the partially filled Fe orbitals at the Fermi level [NiravPRB2016, ; ZhangPRB2017, ], it is natural to believe that the electron-electron (e-e) interactions play a critical role in determining the characteristics of and other FeSCs [Dagotto_mod, ].

To date, theoretical studies of correlated multiorbital models have revealed many new phenomena such as a Hund’s metal [HauleNJP, ; Georges, ; Fanfarillo_PRB2015, ] and the orbital-selective Mott phase (OSMP) [Anisimov, ; Koga, ; deMediciPRL2014, ; Yi_PRL2013, ; ShaozhiPRB2016, ]. These concepts are central to our understanding of the FeSCs. At the same time, however, there is some evidence that the e-ph interaction can also be important in these materials. For example, e-ph interactions have been proposed as pairing mediators in FeSe monolayers grown on oxide substrates [Liu_nature2012, ; Lee_nature2014, ; DHLee2015, ; Xiang2016, ; Rademaker2016, ; KenPRL2017, ], although the idea is under debate [SongPreprint, ; JandkePreprint, ; LiPreprint, ]. Moreover, there is experimental evidence that the low-energy electronic properties of the FeSC are modified by the e-ph interaction, as inferred from Raman [RahlenbeckPRB, ] and infrared spectroscopy [XuPRB2015, ], optical conductivity measurements [DongPRB2010, ], and time-resolved photoemission [GerberScience, ]. On a theoretical front, early ab initio calculations found that the e-ph coupling strength in the FeSCs is minimal, with total dimensionless couplings , and insufficient to establish the observed superconducting transition temperatures [BoeriPRL2008, ]. Several follow-up studies have examined the influence of electronic and magnetic correlations and orbital degrees of freedom on the e-ph interaction in multiorbital systems [SaitoPRB2010, ; YndurainPRB2009, ; LiPreprint, ; Shaozhiarxiv, ; Coh, ; GerberScience, ]. Each has found non-trivial effects and enhanced couplings, indicating that the role of the e-ph interaction may be more subtle than initially expected. Studying e-ph interactions in multiorbital materials is, therefore, an open problem. This issue is also relevant to the quasi-1D organic superconductors of the Bechgaard and Fabre salt families [Dupuis_ltp, ].

A simplified class of models describing the combined e-ph and Coulomb interactions are Hubbard-Holstein (HH) models. In the single-band case, the competition between the local Coulomb repulsion and the attractive e-ph interaction leads to a phase transition between antiferromagnetic and charge-density-wave (CDW) order at half filling [BauerPRB, ; NowadnickPRL, ; MurakamoPRB2013, ; PradhanPRB2015, ; ClayPRL2005, ; HardikarPRB2007, ; KollerPRB2004, ]. This phase transition has been studied extensively in different dimensions [BauerPRB, ; ClayPRL2005, ; NowadnickPRL, ], as a function of doping [ClayPRL2005, ; HardikarPRB2007, ; MendlPRB2017, ], and using many different methods [BauerPRB, ; NowadnickPRL, ]. For example, an early dynamical mean field theory (DMFT) study on the Bethe lattice found this phase transition is continuous for small e-ph couplings and discontinuous for large couplings [BauerPRB, ; JeohPRB, ]. This conclusion differs from that of determinant quantum Monte Carlo (DQMC) calculations on a two-dimensional lattice, which found that the phase transition remains continuous for large couplings at high temperatures [NowadnickPRL, ; JohnstonPRB2013, ]. The phase transition in the one-dimensional HH model is also complicated. No Mott phase transition occurs in the absence of the e-ph interaction, and the Mott insulating phase exists for any onsite Coulomb repulsion [Lieb_PRL1968, ]. Conversely, in the absence of e-e interactions, the critical value of the e-ph coupling strength exists for the formation of a CDW phase due to quantum fluctuations of the phonon field [HirschPRL, ; HirschPRB, ; Bursill_PRL1998, ; Jeckelmann_PRB1999, ]. Retardation effects are also more prominent in one dimension at strong couplings [HirschPRL, ; Fradkin_PRB1983, ] when compared with the infinite-dimensional case [BauerPRB, ]. Thus, while a variety of techniques find a robust phase transition between the CDW and the Mott phases in all dimensions, the details of the transition can differ in important ways. It is therefore important to study HH models in different dimensions and to use different approaches to obtain a complete physical picture.

Another critical factor in the single-band HH model is the degree of electron doping. For example, at half filling, the CDW and antiferromagnetic Mott phases are separated by a metallic phase at weak e-ph coupling on the Bethe lattice [JeohPRB, ]. But at small hole doping, phase separation of the CDW and the metallic phases [CaponePRL2004, ] occurs. In two-dimensions and at a quarter filling, a charge-ordered antiferromagnetic phase appears for strong Coulomb repulsion instead of the usual antiferromagnetic phase [KumarPRB, ]. For quarter filling in 1D, and at weak e-ph coupling, the antiferromagnetic phase separates into an antiferromagnetic phase and a correlated singlet phase [RejaPRB, ]. These results indicate that many new states can arise in the doped single-band HH model. It is then natural to wonder how orbital degrees of freedom enter into this problem.

To date, multiorbital extensions of the HH model have received comparably less attention [KontaniPRL, ; YamadaJPSJ, ; CaponeScience, ; CaponePRL2001, ; HanPRL, ; Shaozhiarxiv, ]. In Ref. [Shaozhiarxiv, ], some of the current authors examined orbital selective behaviors in a degenerate two-orbital HH model (with inequivalent bandwidths) at half filling using single-site DMFT [Shaozhiarxiv, ]. We found that the combined e-ph and onsite e-e interactions resulted in many competing phases including metallic, Mott, and CDW insulating phases, an OSMP, and a lattice-driven orbital-selective Peierls insulator (OSPI) analogue to the OSMP.

Our prior study was carried out in infinite dimensions, where DMFT is exact; however, given the dependence on dimensionality found for the single-orbital HH model, it is essential to study the problem in other dimensions, as a function of doping, and with different techniques. Motivated by this, here we present a complementary study of the three-orbital HH model defined on an extended 1D chain at an average electron filling . We study the model using DQMC, which is a nonperturbative auxiliary-field technique capable of handling both the e-e and e-ph interactions on equal footing.

Figure 1: (color online) A sketch of the - phase diagram for our model as inferred from our DQMC calculations. Four distinct regions are found, which include states with metallic (M) characteristics, an orbital-selective Mott region (OSMP), a charge-density-wave (CDW) order, and a region with strong orbital correlations (OC) and insulating characteristics. The level diagrams sketch the dominant electronic configurations in each region.

Previous studies of this three-orbital 1D Hubbard model found that it exhibits a rich variety of phases including block antiferromagnetism, antiferromagnetism, and several types of OSMPs [RinconPRL, ; LiuPRE, ; ShaozhiPRB2016, ]. As such, the model provides an excellent starting point for studying the impact of the e-ph coupling on such physics. Our results show that the e-ph interaction further modifies the boundaries of these phases by introducing additional competition with the CDW tendencies. By working on an extended lattice, we can study momentum resolved quantities. Our main results are summarized in Fig. 1, which provides a schematic of the phase diagram of the model. (A more precise diagram with quantitative boundaries is provided at the end of our analysis, see Fig. 11.) We find four distinct regions in the - parameter space, where and parameterize the e-ph and e-e interactions, respectively. For small and large we find an OSMP phase consistent with a block antiferromagnetic OSMP reported in prior work [RinconPRL, ; LiuPRE, ]. For large and small we find an insulating CDW phase with mixed ordering. These two phases are separated by a metallic (M) phase at weak and , which persists into the region where and are comparable to one another; however, if these same interactions are made sufficiently large but still comparable, the metallic state develops short-range orbital correlations (OC), resulting in insulating behavior. Our results show that metallicity can be lost in the presence of large e-e and e-ph interactions due to the formation of short-range correlations. We also demonstrate that the addition of e-ph interactions to multiorbital Hubbard models can strongly influence the phases of the model and that this occurs irrespective of the dimension, filling, or mechanism underlying other orbital-selective behaviors.

Ii Methods

ii.1 Model Hamiltonian

Our starting point is a simplified one-dimensional three-orbital Hubbard model, first introduced in Ref. [RinconPRL, ]. We then add a Holstein-type interaction, where the atomic displacement is coupled to the electron density on each orbital. The full Hamiltonian is , where


are the non-interacting electronic terms,


are the non-interacting lattice terms,


are the on-site Hubbard and Hund’s interaction terms, and


are the e-ph coupling terms. Here, denotes a sum over nearest neighbors; () creates (annihilates) a spin electron in orbital 1, 2, 3 on site ; () creates (annihilates) a phonon on lattice site ; is the z-component of the spin operator ; is the number operator; and and are the lattice position and momentum operators, respectively. The parameters are the on-site energies for each orbital; are the intra- and interorbital hopping integrals; and are the intra- and interorbital Hubbard interactions, respectively, and is the Hund’s coupling. The parameter is the strength of the e-ph coupling and is the phonon energy. Finally, is the chemical potential, which fixes the average particle number.

In Eq. (3) we have neglected the pair-hopping and spin-flip terms of the Hund’s interaction, as was done in Ref. [ShaozhiPRB2016, ] for the same model without e-ph interactions. These terms introduce a significant Fermion sign problem [LohPRB, ] for our DQMC calculations and are therefore neglected to make the problem tractable. Prior work [LiuPRE, ] has shown that these terms only change the location of the various phase boundaries for the model considered here in the absence of the e-ph interaction. We, therefore, proceed assuming that this will also hold true once the phonons are included in our calculations.

ii.2 Model parameters

Throughout this work, we choose , as is standard for enforcing rotational symmetry [CastellaniPRL, ], although we have neglected the pair-hopping and spin-flip terms in Eq. (3). We further vary while holding fixed. This choice produces a robust OSMP [RinconPRL, ; LiuPRE, ; ShaozhiPRB2016, ] in the absence of the e-ph interaction and is appropriate for the FeSCs. We work at a fixed filling , which is typical for three-orbital Hubbard models used to describe the 2D FeSCs [DaghoferPRB2010, ]. We expect that the same filling is needed to describe the quasi-one-dimensional system . This choice of filling also allows us to make direct comparisons to previous studies in the absence of the e-ph interaction [RinconPRL, ; LiuPRE, ; ShaozhiPRB2016, ]. In this spirit, we also set eV, eV, eV, eV, eV, eV, eV, and eV, again following Refs. [ShaozhiPRB2016, ] and [RinconPRL, ]. The total bandwidth of the non-interacting model is eV, which serves as the unit energy. The dimensionless e-ph coupling constant is defined as . (Note that since this is a multi-band system, different choices of bandwidths are possible. Here, we select the total bandwidth, as was done in Ref. [Shaozhiarxiv, ].) Finally, we set as units of distance and mass, respectively, and work at an inverse temperature unless stated otherwise; this temperature is low enough to identify the ordering tendencies in the model.

ii.3 Methods

We use DQMC to solve the 1D three-orbital HH model. The general details of the method can be found in the Refs. [BlankenbeclerPRD, ; WhitePRB, ; ChangPMB, ], while the aspects that are unique to the Holstein phonons can be found in the Refs. [JohnstonPRB2013, ] and [ScalettarPRB1989, ]. Throughout this work, we use a one-dimensional chain with a chain size and imaginary time discretization of , unless otherwise stated. In all of our simulations, we have not observed significant errors introduced by this choice.

DQMC calculates the electron Green’s function defined in the imaginary time axis. In Sec. III. B we will examine the spectral properties of our model, which requires an analytic continuation to the real frequency axis. Here, we used the maximum entropy method (MEM) [JarrellPR, ; FuchsPRE, ].

Iii Results

In this study, we will focus on the four phases resulting from the competition between the onsite Coulomb repulsion and the e-ph interaction. The four phases are metal, charge-density-wave phase, orbital-selective Mott phase, and orbital-correlated state, as shown in Fig. 1. They are identified by examining the evolution of the single and double occupations on each orbital, the spectral weight, and the charge-density-wave susceptibility. We will discuss the particular variation of these quantities as a function of parameters in the following sections.

iii.1 Weak Electron-Phonon Coupling

Figure 2: (color online) The variation of electronic densities on the three orbitals as a function of the Hubbard and the e-ph coupling strength . Panels (a) - (c) show the variation of electronic densities as a function of the Hubbard on orbitals , 2, and 3, respectively. Similarly, panels (d) - (f) show the change of electronic densities as a function of on the same three orbitals. In each panel, error bars smaller than the marker size have been suppressed for clarity, and a smoothing spline is used as a guide to the eye.
Figure 3: (color online) The variation of double occupancies on the three orbitals as a function of the Hubbard and the e-ph coupling strengths . Panel (a) - (c) shows the variation of double occupancies as a function of the Hubbard on orbitals , 2, and 3, respectively. Similarly, panels (d) - (f) show the change of double occupancies as a function of on the same three orbitals. In each panel, error bars smaller than the marker size have been suppressed for clarity, and a smoothing spline is used as a guide to the eye.
Figure 4: (color online) Momentum dependence of the charge-density-correlation for orbital 1 (a), orbital 2 (b), and orbital 3 (c) at different values. The Hubbard . In each panel, error bars smaller than the marker size have been suppressed for clarity.
Figure 5: (color online) Momentum dependence of the charge-density-correlation for orbital 1 (a), orbital 2 (b), and orbital 3 (c) at different values. The Hubbard and . The black dashed lines are eye-guided lines for a combination of two Lorentzian functions with different peak positions. The two Lorentzian functions are shown with dotted and dash-dotted lines, respectively. In each panel, error bars smaller than the marker size have been suppressed for clarity.
Figure 6: (color online) The variation of the spectral weights for three orbitals as a function of the Hubbard and the e-ph coupling strength . (a) - (c) show the variation of spectral weights as a function of the Hubbard for three orbitals, respectively. (d) - (f) show the change of spectral weights as a function of for three orbitals, respectively. In each panel, error bars smaller than the marker size have been suppressed for clarity, and a smoothing spline is used as a guideline to the eye.
Figure 7: (color online) Spectral functions for and . (b),(c), and (d) are the orbital 1, 2, and 3 parts of the spectral function in (a), respectively. (e), (f), (i), and (j) are spectral functions of the orbital 1 at momentum , , , and , respectively. Similarly, (g), (h), (k), and (l) are spectral functions of the orbital 2 at those four momenta. The black dashed lines show three peaks positions in the maximum entropy results. The red dotted lines are Lorentzian fitting results.

Previous studies of this model without phonons showed that an OSMP forms for values in the range for our choice of . In this state, orbital three becomes insulating while the remaining orbitals host itinerant electrons [RinconPRL, ; LiuPRE, ]. The onset of this phase is signaled by the fact that the filling on orbital three . For , orbitals one and two retain a noninteger filling but are driven into an insulating state by the onset of short-range orbital ordering [ShaozhiPRB2016, ]. To avoid this complication, we restrict ourselves to the region.

We now examine the impact of the e-ph interaction on the OSMP. Figure 2 plots the electronic occupations of the three orbitals for different values of and . For , Fig. 2(c) shows that converges to 1 as increases, implying that a Mott gap is formed on this orbital for . At the same time, and maintain noninteger values, implying that these orbitals remain itinerant [see Fig. 2(a) and 2(b)]. These results are consistent with previous studies [RinconPRL, ; LiuPRE, ]. When we include the e-ph coupling, the orbital occupations are modified significantly. For example, Figs. 2(d) - 2(f) show that the e-ph coupling tends to make electronic occupations on all three orbitals uniform when , with the average filling on each orbital approaching when is large. This value of the occupation on each orbital is consistent with a charge-ordered state where two sites are fully occupied, and one site is empty, which is shown in the CDW region of Fig. 1. This kind of charge order arises from the attractive interaction mediated by the e-ph interaction [PhilippePRB, ]. For the fully occupied site, the attractive interaction can be mapped into a negative effective for all three orbitals at this site in the large limit. For an empty site, the effective interaction is not modified by the e-ph coupling. Therefore, the effective Hubbard interaction is not uniform in the real space if there is a density modulation. When the e-ph coupling is strong, this nonuniform attractive interaction can produce a charge-ordered state but with a uniform average occupation on each orbital. The transition from a CDW phase to an OSMP can be seen in Fig. 2(c), where decreases from to 1 at . Increasing the e-ph coupling pushes this transition to larger values of ; for example, for it occurs at .

The competition between the CDW and OSMP tendencies is also manifest in the behavior of the orbital’s double occupation , as summarized in Fig. 3. When the phonon mediated effective attraction overcomes the Coulomb repulsion we expect ; otherwise, . Figures 3(a)-3(c) present as a function of for fixed values of , where we find that decreases as is increased, and converges to in the limit of a strong Hubbard interaction, consistent with a Mott insulating state where double occupation is suppressed. Figures 3(d)-3(f) alternatively plot the data as a function of for fixed values of . Here, we find that for , increases as increases and converges to on each orbital. This value is consistent with the double occupations expected for the CDW phase shown in Fig. 1.

The electronic density and double occupations provide indirect evidence of the CDW phase. To obtain more direct evidence of a CDW order, we calculated the charge susceptibility


where is the momentum, is the imaginary time, , and is a lattice vector.

Figure 4 shows the momentum dependence of the three intraorbital charge susceptibilities for and different e-ph coupling strengths. At weak coupling (i.e. and ), is small, with no clear peak at any momenta. This observation implies that a finite value of is needed for charge correlations to develop at this temperature, and is consistent with the one-dimensional Holstein model [HardikarPRB2007, ]. As the value of is increased, a clear peak structure forms in . For instance, already at we find a peak centered at for all three orbitals, indicating the formation of a two-sublattice charge correlation at . Upon further cooling of the system, we find that additional charge correlations develop at a second point. For example, Fig. 5 compares at and for the and case. For temperature , has a single peak at ; however, as the temperature is decreased to , increases and a second peak forms at , which is evident as a shoulder in . To better recognize these two peaks, we plot as a guide-to-the-eye the sum of two Lorentzian functions centered at (dash-dot line) and (dotted line).

The two peak structures in likely reflect different ordering tendencies. The charge configurations sketched in Fig. 1 are consistent with and orderings. We, therefore, propose that the CDW state is characterized by a superposition of and configurations along the chain, where the number indicates the number of carriers on each site. These charge configurations are consistent with the values of the orbitally-resolved single and double occupancies discussed previously. In fact, these two peaks reflect two different values of that appear in this multiorbital model; The Fermi momentum for orbitals 1 and 2 are , while the Fermi momentum for orbital 3 is . Thus, these two peak values correspond to in the weak coupling limit, where the CDW tendencies are driven primarily by nesting conditions. When we increase the e-ph coupling further, the for orbitals 1 and 2 increases to . Therefore, we expect that only one peak will be observed in the charge-density-wave susceptibility in the limit of strong e-ph coupling.

We now turn to the spectral weight of the three orbitals in the vicinity of the Fermi level to assess whether the various phases we observe are insulating or not. The spectral weight can be estimated directly from the imaginary time Green’s function using the relationship [TrivediPRL, ]


where is the orbitally-resolved spectral function. At low temperature, the function is sharply peaked around and thus provides a measure of the spectral weight integrated over a window of a few of .

Figure 6 plots for the three orbitals for different values of and . In Figs. 6(a)-6(c), the spectral weight of each orbital is plotted as a function of for fixed . In the absence of the e-ph interaction (), the spectral weight of all three orbitals decreases as the Hubbard increases. However, the spectral weight on orbital decreases more rapidly than the other two orbitals, consistent with the formation of an OSMP [ShaozhiPRB2016, ]. [The small but nonzero value of is due to the elevated temperature of the simulation.] As the value of increases, we begin to see the loss of spectral weight in all three orbitals when is small.

Fig. 6(d)-6(f) plot as a function of for fixed . For , the spectral weight of all three orbitals is suppressed as the e-ph coupling is increased. We further observe a sudden decrease in the spectral weight of all three orbitals for , where a prominent peak in is observed. Thus, for and , the system is an insulating state driven by CDW correlations. The behavior of the spectral weight is qualitatively similar for , however, the transition to the CDW phase occurs at larger values of as increases. Based on these results, we conclude that the CDW phase is insulating.

Unlike the OSMP, we do not find any orbital-selective behavior associated with the formation of the CDW phase; the rate at which the spectral weight approaches zero appears to be the same for all three orbitals at this temperature. This result is in contrast to the degenerate two-orbital case with inequivalent bandwidths [Shaozhiarxiv, ], where orbital selective CDW behavior was found. This difference could be attributed to changes in the total bandwidth, dimensionality, or model. (For example, the current model has inequivalent bandwidths and crystal field splittings while the former only had inequivalent bandwidths.) Further studies will be needed to better understand the differences between these two cases.

We now return to the competition between the OSMP and CDW phases. Fig. 6(a)-(c) reveals that the spectral weight decreases as the strength of the e-ph is increased when is small. For a fixed value of , initially increases with , before reaching a maximum value and decreases along the axis. This behavior reflects the competition between the e-ph and the e-e interactions [NowadnickPRL, ; NowadnickPRB, ]. At small the CDW correlations dominate, for large the OSMP correlations dominate, and for intermediate values of a metallic phase is realized.

iii.2 Spectral Properties of the CDW Phase

Figure 8: (color online) Spectral functions for and . (b), (c), and (d) are the orbital 1, 2, and 3 parts of the spectral function in (a). The inverse temperature is . The white dot line is the Fermi surface.

The spectral function of the OSMP was studied in detail in Ref. [ShaozhiPRB2016, ] in the absence of e-ph interactions. We will, therefore, focus on the spectral function of the CDW phase. Figure 7 shows the spectral function for and and its orbitally-resolved components. The system is insulating, with a large CDW gap and broadened spectral features, consistent with our spectral weight analysis. The upper bands of three orbitals have dispersions with a clear folded shape, while the lower bands of orbitals 1 and 2 have a more cosine-like dispersion. This cosine-like shape arises from the combination of an incoherent peak and an additional peak arising from thermally activated transitions to states with additional phonons excited [HohenadlerPRB, ]. To better recognize these two peaks, Figs. 7(e) - 7(l) plot the spectral functions at fixed momentum , , , and for orbitals 1 and 2. The red dashed curve are Lorentzian fits of the data allowing for an incoherent peak above and below the Fermi level and an additional thermally excited peak below the Fermi level. The fitting results are consistent with the Maxent results. We find that the thermally excited state is located around eV and is momentum independent, consistent with previous results for the one-dimensional single-band spinless Holstein model [HohenadlerPRB, ]. The folded band is observed at and and at and , respectively. The intensity of the incoherent peak below the Fermi level in the folded band is much weaker than that of the thermally excited peak, leading to a cosine shape observed in the upper panels of Fig. 7.

iii.3 Strong electron-phonon coupling

Figure 9: (color online) Cartoon sketch of the relevant charge fluctuation processes leading to the orbital ordered insulating state when and .

Our previous single-site DMFT study of the two-orbital HH model at a half filling observed a direct transition between the OSMP and CDW phase in the strong e-ph coupling limit [Shaozhiarxiv, ], with no intervening metallic phase. In contrast, for the current model, we find evidence for an orbitally correlated insulating state located between the CDW phase and OSMP at strong couplings. Figure 8 shows the spectral function [Fig. 8(a)] and its three orbitally-resolved components [Figs. 8(b) - 8(d)] at , , and , where a gap is clearly observed. The spectral function is similar to the CDW case shown in Fig. 8, however, the origin of the gap is not CDW correlations since (not shown) is small in this state. Also, the double occupation (see Fig. 2), indicating the Coulomb interaction is the dominant interaction in this phase.

The nature of this OC phase is sketched on the left side of Fig. 9. It consists of one site where all three orbitals are fully occupied and two neighboring sites that are half filled and in a high-spin state. This electronic configuration is consistent with the observed orbital occupations and the value of the double occupation .

In the HH model, the intra- and interorbital Coulomb interactions and are renormalized by the e-ph interaction, and the ground state can change based on the value of these effective Coulomb interactions. Figures 9(a) and 9(b) show two types of charge fluctuations that are possible within the proposed OC state. The potential energy cost of these fluctuations are and , respectively. To estimate the magnitude of these energies, we performed an exact diagonalization calculation in the atomic limit and compared the ground state energies of the shown atomic configurations. For and , we find that these two potential energies are 1.305 eV and 0.815 eV, respectively. When the orbital hybridization is introduced, the total potential energy cost is compensated for by a kinetic energy gain of  eV. However, the ratio in both cases, suggesting that charge fluctuations are suppressed, and the system will be insulating. The conditions for forming the OC insulating state are then and , which in turn requires that the e-ph coupling strength is not too strong; otherwise, the CDW phase is formed. (Note that a larger Hund’s coupling favors satisfying these two conditions.) In the OC insulating state, the fully occupied site and two half occupied sites can be arranged randomly in a long chain as the energy cost will not change. Therefore, short-range orbital correlations would be sufficient to produce insulating behavior at finite temperature.

Figure 10: (color online) (a) The electronic occupations of orbital as a function of the e-ph coupling strength . (b) The onsite orbital correlations between orbital and as a function of . These results were obtained on an site chain. In each panel, error bars smaller than the marker size have been suppressed for clarity.

We can confirm the presence of OC by examining the equal time orbital correlation function


The charge configuration expected for the OC shown in Fig. 9 would produce an orbital correlation function . Figure 10(b) shows the variation of the onsite orbital correlations in the phase transition from the OSMP to the OC state. In general, the correlation function for is larger than that for . Thus, the onsite correlation is a good indicator for when and can be used to trace the forming of the orbital correlations. In the OSMP region, is greater than 0.1. For example, at , , , and are 0.16, 0.14, and 0.125, respectively. decreases slowly initial as the e-ph coupling strength is enhanced. Conversely, near the phase transition, decreases very quickly; at , , , and are 0.056, 0.061, and 0.057, respectively. The nonzero value at is likely due to the elevated temperature. We find , , and are decreased to 0.038, 0.03, and 0.029, respectively, as the inverse temperature is decreased to . We expect that the correlation function tends towards zero as , and a sharp phase transition from the OSMP to the OC state would occur.

We traced the phase transition from the OSMP to the OC in Fig. 10(a), which shows the variation of orbital occupations in the phase transition from the OSMP to the OC phase at and . Here, the chain size is . The critical e-ph coupling value of the phase transition from the Mott phase to the OC state is about 0.43, where . As and , orbital occupations for three orbitals are 1.4, 1.38, and 1.22 at , respectively. Those occupations are changed to 1.346, 1.344, 1.31 at , implying the OC state supports the same occupation on each orbital, consistent with the electron configuration shown in Fig. 9.

Iv Discussion And Summary

Figure 11: (color online) The phase diagram of the three-orbital Hubbard-Holstein model for and . The different phases are labeled as follows: metal (M), charge-density-wave order (CDW), orbital selective Mott phase (OSMP), and orbital correlation(OC).

We have performed a study of a three-orbital Hubbard-Holstein model on an extended one-dimensional chain using non-perturbative DQMC. The phase diagram of the one-dimensional model for and , shown in Fig. 11, shares many similarities to the one found for an infinite dimensional degenerate two-orbital HH model with inequivalent bandwidths [Shaozhiarxiv, ], containing a metallic phase, a CDW phase, and an OSMP. The metallic phase is most prominent in small values of and but penetrates into the region of intermediate interaction strengths separating the CDW and OSMP when . The critical e-ph coupling needed for the CDW phase transition at in our model is . The nonzero value of is consistent with results for the single-band Holstein model [HirschPRL, ; HardikarPRB2007, ].

At strong couplings, we found evidence for an orbital correlation state in the phase diagram, which was not found in the previous DMFT study. We argue that this difference stems from the filling used in our model ( here versus in Ref. [Shaozhiarxiv, ]) and the use of an extended cluster here [ShaozhiPRB2016, ]. The OC state resides between the CDW phase and OSMP and tends to extend to large Hubbard . This region of the phase diagram is the same one where the OSMP disappears, and an anti-ferro-orbital correlation was found in the case [ShaozhiPRB2016, ]. We expect that a phase transition occurs between the anti-ferro-orbital order and the OC state at . These results show that the phase diagram of multiorbital HH models can exhibit remarkably rich physics as a function of interaction strengths, doping, and other parameters.

In the past, the e-ph interaction has been neglected when studying the FeSCs. As mentioned in the introduction, this is largely motivated by early ab initio calculation showing that the dimensionless interaction strength is only about 0.2 [BoeriPRL2008, ]. However, we find the electronic degrees of freedom in our 1D multiorbital model can be influenced significantly by small e-ph couplings. Given that a similar result was obtained in the case of a two-orbital Hubbard model in infinite dimensions using DMFT [Shaozhiarxiv, ], these results imply that the e-ph interaction cannot be neglected a priori in multiorbital systems, and that this conclusion holds independent of the dimension of the system. We hope that this study will motivate further work in this interesting area.

V Acknowledgements

This work was supported by the Scientific Discovery through Advanced Computing (SciDAC) program funded by U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research and Basic Energy Sciences, Division of Materials Sciences and Engineering. CPU time was provided by resources supported by the University of Tennessee and Oak Ridge National Laboratory Joint Institute for Computational Sciences (


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