Abstract
The electron–electron and electron–phonon interactions play an important role in correlated materials, being key features for spin, charge and pair correlations. Thus, here we investigate their effects in strongly correlated systems by performing unbiased quantum Monte Carlo simulations in the square lattice HubbardHolstein model at halffilling. We study the competition and interplay between antiferromagnetism (AFM) and chargedensity wave (CDW), establishing its very rich phase diagram. In the region between AFM and CDW phases, we have found an enhancement of superconducting pairing correlations, favouring (nonlocal) swave pairs. Our study sheds light over past inconsistencies in the literature, in particular the emergence of CDW in the pure Holstein model case.
Introduction
The electron–phonon (eph) interaction is a central issue in condensed matter, in particular when discussing properties of conventional superconductivity (SC) and charge ordering^{1}. While Bardeen, Cooper and Schrieffer used this interaction in their seminal work to explain pairing^{2}, Peierls took it into account to provide a mechanism, based on Fermi surface nesting (FSN), that leads to chargedensity wave (CDW)^{3}. Recently, the debate about the role of the eph coupling has been intensified due to the occurrence of unconventional (non Peierlslike) CDW phases, and their competition with SC, in some classes of materials, such as transitionmetal dichalcogenides^{4,5,6,7}. Even for cuprates, materials known by their strong electron–electron (ee) interactions, recent findings provided evidence for the occurrence of CDW in the doped region, with competing effects with SC^{7,8,9,10}, e.g., on doped La_{2−x}Ba_{x}CuO_{4} and YBa_{2}Cu_{3}O_{6+x}^{11,12,13,14}. These results have suggested that the phase diagram of highT_{c} superconductors^{15} is far more complex than previously supposed, and have raised issues about the relevance of the eph coupling for correlated materials, rather than just ee interactions.
From a theoretical point of view, a simplified Hamiltonian that captures the interplay between antiferromagnetism (AFM), CDW, and SC is the singleband HubbardHolstein model (HHM)^{16}. It exhibits Coulomb repulsion between electrons, leading to spin fluctuations; and also electronion interactions, which enhance charge/pairing correlations. The emergence of longrange order depends on the competition between these tendencies. This model was vastly studied in onedimensional systems, with wellknown phase diagrams presenting spindensity wave, bondorderwave, CDW, and also metallic or phase separation behavior^{17,18,19,20,21,22,23,24}. A remarkable feature in 1D systems is the occurrence of a quantum phase transition from a metallic LutherEmery liquid phase^{25} to a CDW insulator at a finite critical eph coupling, in the limit case of the pure Holstein model (U = 0), despite of the FSN^{26,27}. By contrast, the properties of the HHM in twodimensional systems are not entirely clear, even for simple geometries, such as the square lattice. For instance, the existence of such a metalCDW quantum critical point (QCP) is matter of controversies for the pure Holstein model in 2D lattices^{28,29,30,31,32,33}. The scenario is much less clear in presence of a repulsive Hubbard term (U ≠ 0), in spite of the large effort to characterize the model^{16,31,32,33,34,35,36,37,38,39,40,41,42,43,44}, since no unbiased results are available for quantum AFM/CDW transitions, to the best of our knowledge.
In view of these open issues, and as a step towards a better understanding of the role of eph interactions in strongly interacting systems, we investigate in this article the competition between AFM and CDW in the square lattice HHM at halffilling, as well as its pairing response, using unbiased quantum Monte Carlo (QMC) simulations. We determine precise critical points for the HHM at intermediate interaction strengths, presenting benchmarks for lattices with linear size up to L = 48 (i.e., 2304 sites) in some cases. Our main results are summarized in the ground state phase diagram displayed in Fig. 1. Here we highlight [i] the absence of a finite critical eph coupling for the pure Holstein model, i.e., the CDW phase sets in for any λ > 0 (and U = 0); [ii] the existence of a finite AFM critical point on the line U = λ, which is strongly dependent on the phonon frequency; and [iii] an enhancement of nonlocal swave pairing in the region between the AFM and CDW phases. These results are also compared with other methodological approaches, such as variational QMC.
Results
The model
The HubbardHolstein Hamiltonian reads
where \({d}_{{\bf{i}}\sigma }^{\dagger }\) (\({d}_{{\bf{i}}\sigma }\)) is a creation (annihilation) operator of electrons with spin σ( = ↑, ↓) at a given site i on a twodimensional square lattice under periodic boundary conditions, with 〈i, j〉 denoting nearestneighbors, and \({n}_{{\bf{i}}\sigma }\equiv {d}_{{\bf{i}}\sigma }^{\dagger }{d}_{{\bf{i}}\sigma }\) being number operators. The first two terms on the right hand side of Eq. (1) correspond to the kinetic energy of electrons, and their chemical potential (μ) term, respectively, while the onsite Coulomb repulsion between electrons is included by the third term. The ions’ phonon modes are described in the fourth term, in which \({\hat{P}}_{{\bf{i}}}\) and \({\hat{X}}_{{\bf{i}}}\) are momentum and position operators, respectively, of local quantum harmonic oscillators with frequency ω_{0}. The last term corresponds to local electronion interactions, with strength g. Hereafter, we define the mass of the ions (M) and the lattice constant as unity.
It is also worth to introduce additional parameters, due to the effects of the phonon fields to the electronic interactions. From a second order perturbation theory on the eph term^{16}, one obtains an effective dynamic ee interaction, \({U}_{{\rm{eff}}}(\omega )=U\frac{{g}^{2}/{\omega }_{0}^{2}}{1{(\omega /{\omega }_{0})}^{2}}\), with \({g}^{2}/{\omega }_{0}^{2}\equiv \lambda\) being the energy scale for polaron formation. The appearance of such a retarded attractive interaction, depending on the phonon frequency, leads us to define ω_{0}/t as the adiabaticity ratio, and λ/t as the strength of the eph interaction. To facilitate the following discussion, we also define U_{eff} ≡ U − λ, which gives us a rough information about the local effective ee interaction, and is also an important parameter to our methodological approaches. Furthermore, we set the electron density at halffilling, i.e., 〈n_{iσ}〉 = 1/2.
We investigate the properties of Eq. (1) by performing two different unbiased auxiliaryfield QMC approaches: the projective ground state auxiliaryfield (AFQMC)^{45,46}, and the finite temperature determinant quantum Monte Carlo (DQMC) methods^{46,47,48}. Following the procedures described in Karakuzu et al.^{49}, we implemented a signfree AFQMC approach to the halffilling of the HHM, allowing us to analyze large lattice sizes, but, conversely, being restricted to the U_{eff} ≥ 0 region. The properties of the U_{eff} < 0 region, forbidden to our AFQMC method, are investigated by DQMC simulations. We recall that the DQMC method may exhibit sign problem for the HHM, depending on the strength of parameters. However, as shown in the Supplementary Figure 4, the values for the average sign with U < λ are still suitable for performing accurate simulations for the CDW phase, in particular for intermediate interaction strengths, allowing us to obtain the physical quantities of interest, in some cases up to L = 14 and β = 1/T = 28. In fact, the DQMC average sign is strongly suppressed just around U ≈ λ, where our signfree AFQMC approach works. Thus, our AFQMC and DQMC simulations are used complementarily.
The charge and magnetic responses are quantified by their respective structure factors, i.e., \({S}_{{\rm{cdw}}}({\bf{q}})=\frac{1}{N}{\sum }_{{\bf{i}},{\bf{j}}}{e}^{{\rm{i}}{\bf{q}}\cdot ({\bf{i}}{\bf{j}})}\langle {n}_{{\bf{i}}}{n}_{{\bf{j}}}\rangle\), and \({S}_{{\rm{afm}}}({\bf{q}})=\frac{1}{N}{\sum }_{{\bf{i}},{\bf{j}}}{e}^{{\rm{i}}{\bf{q}}\cdot ({\bf{i}}{\bf{j}})}\langle {S}_{{\bf{i}}}^{z}{S}_{{\bf{j}}}^{z}\rangle\), with n_{i} = n_{i↑} + n_{i↓}, \({S}_{{\bf{i}}}^{z}={n}_{{\bf{i}}\uparrow }{n}_{{\bf{i}}\downarrow }\), and N = L × L being the number of sites. This allows us to probe their critical behavior by means of the correlation ratio
with ∣δq∣ = 2π/L, q = (π, π), and ν ≡ cdw or afm. According to well established finite size scaling analysis, the critical region is determined by the crossing points of R_{ν}(L) for different lattice sizes^{50,51,52,53}. Finally, the pairing properties are investigated by the finite temperature superconducting pair susceptibility \({\chi }_{{\rm{sc}}}^{\alpha }(\beta )=\frac{1}{N}\mathop{\int}\nolimits_{0}^{\beta }{\rm{d}}\tau \ \langle {\Delta }_{\alpha }(\tau ){\Delta }_{\alpha }^{\dagger }(0)\rangle ,\) with \({\Delta }_{\alpha }(\tau )=\frac{1}{2}{\sum }_{{\bf{i}},{\bf{a}}}{f}_{\alpha }({\bf{a}}){c}_{{\bf{i}}\downarrow }^{\dagger }(\tau ){c}_{{\bf{i}}+{\bf{a}}\uparrow }^{\dagger }(\tau )\), \({c}_{{\bf{i}}\sigma }(\tau )={e}^{\tau {\mathcal{H}}}{c}_{{\bf{i}}\sigma }{e}^{\tau {\mathcal{H}}}\) and f_{α}(a) being the pairing form factor for a given symmetry. Here, we consider onsite, nearestneighbors (NN), and nextnearestneighbors (NNN) spinsinglet pairing operators for the swave symmetry, which are denoted by α ≡ s, s*, and s** (or s_{xy}, as also called in literature), respectively; and also consider the \({d}_{{x}^{2}{y}^{2}}\)wave symmetry, α ≡ d (see, e.g., White et al.^{54}).
The Holstein model (U = 0)
We first discuss the limit case of U = 0 in Eq. (1), i.e., the pure Holstein model. While the Peierls’ argument^{55} suggests an insulating charge ordered ground state for any λ > 0, onedimensional systems exhibit a metalCDW transition for a finite critical λ_{c}, despite the perfect FSN. In two dimensions, the occurrence of a finite critical λ_{c} in the square lattice is controversial: while variational QMC approaches provide evidence for λ_{c}/t ≈ 0.8^{32,33}, unbiased QMC results suggest the nonexistence of a finite critical point^{30,31}. Here, we address this controversy by analyzing the critical behavior given by the CDW correlation ratio, Eq. (2), by means of DQMC simulations One should notice that, for U = 0, the DQMC method does not suffer with the sign problem, allowing us to investigate larger lattice sizes and lower temperatures.
The quantum critical region is probed by the behavior of the correlation ratio as a function of the eph coupling, and projecting on the ground state by using β ~ L, for different system sizes. Fig. 2 displays the R_{cdw}(L) as a function of λ/t for several lattice sizes, fixing U = 0 and β = 2L. Here, we start discussing the possible existence of a CDW quantum phase transition at λ_{c}/t ≈ 0.8, as suggested by variational methods. By fixing λ/t = 0.8, we notice that the correlation ratio increases monotonically as a function of the lattice size, clearly supporting longrange CDW order in the thermodynamic limit, at this interaction strength and above, in line with Hohenadler et al.^{30}. In fact, one may estimate the QCP by the value of λ/t determined by a crossing of R_{cdw}(L) for different system sizes. Here we define λ_{c}(L, L − ΔL) as the sizedependent critical coupling obtained by the crossing between R_{cdw}(L) and R_{cdw}(L − ΔL). As presented in the inset of Fig. 2, the values of λ_{c}(L, L − ΔL) are smaller than the one suggested by the variational methods, and they also are reduced when L increases. Given this, whether a metalinsulator transition exists, it should occur at smaller coupling strengths.
A thorough determination of the existence of a critical point is given by a finite size scaling analysis of λ_{c}(L, L − ΔL). Following the procedure adopted in Weber et al.^{31}, which hereafter is used to determine the CDW transitions, we perform a power law fit [f(L) = a + bL^{c}] of the crossing points, as displayed in the inset of Fig. 2. Within this scaling analysis, the critical coupling (at L → ∞) λ_{c} for U = 0 is consistent with a vanishing or very small value, even when we adopt the less accurate (2nd order) polynomial fit. Similar analysis for a different ground state projection, i.e. β ~ L^{2}, also agrees with a vanishing λ_{c} for U = 0; see also the Supplementary Notes 1. That is, these results provide evidence that a finite critical eph coupling is not plausible for the square lattice Holstein model. The difference between the square lattice and onedimensional systems may stem on the larger electronic susceptibility of the former, which diverges with the square logarithm of temperature.
The HubbardHolstein model
We now turn to discuss the behavior of the HHM for U ≠ 0, investigating initially the particular case of U = λ, by means of AFQMC simulations. We recall that the HHM in the ω_{0} → ∞ limit is equivalent to the Hubbard model with an onsite interaction U_{eff}. Then, it should exhibit a metallic behavior along the line U = λ in this case. However, the existence of finite phonon frequencies leads to a retarded interaction, therefore to a more complex ground state. Indeed, our AFQMC results for ω_{0}/t ≤ 1 exhibit an enhancement of the spinspin correlations as a function of U/t, on the line U = λ, as showed in the Supplementary Fig. 1. Conversely, the chargecharge response remains weak for any interaction strength, suggesting that an AFM order sets in at ground state.
Similarly to the previous CDW analysis, the AFM longrange order is established by investigating the crossing points of the AFM correlation ratio. Fig. 3 displays the U_{c}(L, αL), i.e. the values of U/t for the crossing points of R_{afm}(L) and R_{afm}(αL), for different phonon frequencies, and fixing β = 2L. Due to the large lattice sizes achieved in our AFQMC simulations, here we adopt a linear polynomial fit for the AFM transitions. As expected, the pure Hubbard model (black square symbols in Fig. 3) is AFM for any U > 0, i.e., U_{c} = 0. However, in presence of eph coupling (along the line U = λ), a quantum phase transition occurs, changing from a correlated metalliclike ground state to an ordered AFM one, for a given coupling strength. Dynamical meanfield theory analyses also report similar results for higher dimensionality calculations^{36,37,38}. In our QMC simulations, for instance, one finds U_{c}/t = λ_{c}/t = 0.73(6) by fixing ω_{0}/t = 1; that is, the ground state is AFM for any U = λ > 0.73t. The position of this QCP strongly depends on the choice of ω_{0}/t, as showed in the inset of Fig. 3. A similar analysis for a different ground state projection (β ~ L^{2}) agrees with it, but suggesting slightly larger critical couplings, as presented in the Supplementary Notes . Notice that such increasing behavior of U_{c} as a function of ω_{0} is consistent with the initial expectation of an emergent metallic behavior in the antiadiabatic limit for U = λ. The properties of this correlated metalliclike state are discussed later.
We proceed by investigating the quantum critical behavior for the general case, U ≠ λ. To this end, here we analyze the AFM and CDW correlation ratios as functions of U/t, for fixed λ/t and ω_{0}/t, e.g., as represented by the vertical dotted line in Fig. 1. With this in mind, Fig. 4a displays AFQMC results for R_{afm}(L) as function of U/t, and fixed λ/t = 2 and ω_{0}/t = 1. The crossing points U_{c}(L, L − ΔL) between R_{afm}(L) and R_{afm}(L − ΔL), as well as their finite size scaling, are displayed in Fig. 4c, leading to an AFM quantum phase transition at \({U}_{c}^{{\rm{AFM}}}/t=1.88(2)\). Similarly, the QCP for a CDW transition may be obtained by DQMC simulations of R_{cdw}(L), as presented in Fig. 4b, with crossing points and finite size scaling shown in Fig. 4c, leading to \({U}_{c}^{{\rm{CDW}}}/t=1.63(1)\).
When the above analysis is repeated for other values of λ/t, we obtain the phase diagram presented in Fig. 1, with the QCPs being reported in the Supplementary Tables. It is worth mentioning that, for the range of parameters analyzed, we obtain continuous transitions for both AFM and CDW phases, without coexistence, and with a metalliclike (or SC) region between them. First order transitions may occur for stronger coupling, as suggested in the recent literature^{32,33}.
Pairing susceptibility response
Finally, it is instructive to discuss the properties of the region between AFM and CDW phases, from which it is expected the emergence of SC. Since our DQMC method exhibits a small average sign when U ≈ λ, then establishing longrange SC order is challenging, and we are restricted to smaller lattice sizes and high temperatures; see, e.g., Supplementary Fig. 5. Despite this, one is able to investigate the SC properties by analyzing the tendency of the pairing susceptibility as a function of the temperature. For instance, Fig. 5a displays the behavior of χ_{α(sc)} for fixed λ/t = 2, and U/t = 1.7. Here, we notice an enhancement in the pairing correlations at low temperature, with the dominant symmetry being the dwave^{41,44}.
However, a better estimation for pairing is given by extracting the particleparticle contribution of χ_{α(sc)}. Thus, here we define the effective pair (vertex) susceptibility^{54} as \({\chi }_{\alpha {\rm{(sc)}}}^{{\rm{eff}}}={\chi }_{\alpha {\rm{(sc)}}}{\bar{\chi }}_{\alpha {\rm{(sc)}}}\), with \({\bar{\chi }}_{\alpha {\rm{(sc)}}}\) being the noninteracting susceptibility. A positive (negative) response of \({\chi }_{\alpha {\rm{(sc)}}}^{{\rm{eff}}}\) corresponds to an enhancement (weakening) of an attractive pair channel for the αth symmetry. Fig. 5b exhibits \({\chi }_{\alpha {\rm{(sc)}}}^{{\rm{eff}}}\) for the data of panel (a), showing negative tendency towards all of the examined channels. In particular, the onsite swave has the largest negative response, which shows the harmfulness of the Hubbardlike term for local pairs formation. However, since T_{sc} ~ ω_{0} from the BCS theory^{2}, further insights about the nature of this region may be given by increasing ω_{0}, while keeping U_{eff} fixed, as displayed in Fig. 5c, d, for ω_{0}/t = 4, λ/t = 2, and U/t = 1.7. For these parameters, despite the increasing dominant behavior of χ_{α(sc)} for the dwave, only the NNN swave exhibits a positive effective susceptibility. Thus, these combined features suggest that, whether SC emerges at ground state, it likely is (nonlocal) swave.
Indeed, since this metalliclike region seems to be in the negative U_{eff} side of the phase diagram in Fig. 1, the swave symmetry is naively expected. Interestingly, such an enhancement of nonlocal swave response suggests that shortrange charge/spin correlations may suppress the formation of local (swave) and NN (s*wave) Cooper pairs, making the NNN ones (s**wave) the main channel for pairing. It is also important to notice that, as a consequence of the KohnLuttinger weak coupling argument^{56}, instabilities in the particleparticle channel are expected in this intermediate region without AFM and CDW orders, which could lead to pairing^{57,58,59}. However, these instabilities are believed to occur at very low temperatures, usually not accessible for the current QMC methodologies. Finally, we warn that a more precise determination of the nature of this region may also require the analysis of dynamical quantities, or longrange effective electronic interactions^{43,60}.
Discussion
We have presented results for the HHM in the square lattice, using unbiased AFQMC and DQMC methods complementarily, which provide a broader picture about the physical responses of this model. In particular, we have shown that, different from onedimensional systems, the emergence of the CDW phase in the square lattice occurs for any λ > 0, for U = 0. However, these CDW correlations are strongly affected by a Coulomb interaction (U ≠ 0), with the occurrence of AFM even at U_{eff} < 0. We also observed the existence of a correlated metalliclike region between CDW and AFM phases, with an enhancement of nonlocal swave pairing, rather than dwave. Despite the difficulty to establish longrange order for SC, one may expect that swave SC sets in at zero temperature.
These findings constitute a significant step towards a better understanding of this model, by providing precise QCPs, and shedding lights over past theoretical inconsistencies, although there still remain open questions, such as the behavior of the phase boundaries as a function of ω_{0}. Further investigations on a given compound may require the inclusion of its key features. In the cuprates, e.g., the electronphonon coupling effects are due to the outofphase vibrations of planar oxygens along the caxis, the B_{1g} phonon mode, which is momentum dependent (i.e., it is not Holsteinlike), and favors dwave symmetry^{61,62}. However, doing this would also demand an increasing degree of complexity for the models considered (multiband systems, nonHolstein phonon modes, etc)^{63,64}, and also a reduction in the energy scale of the phonon fields, since ω(q) << t for most of the realistic materials. Apart from these specificities, the results for the HHM emphasize the fundamental role of the eph interaction in strongly correlated systems, which is important to the emergence of charge order and pairing, and may be relevant for the physics of novel correlated materials.
Methods
Quantum Monte Carlo simulations
We employed two different approaches: the DQMC and the AFQMC methods. Briefly, the DQMC (AFQMC) approach is based on the decoupling of the noncommuting terms of the Hamiltonian in the partition function (projection operator) by TrotterSuzuki decomposition, which discretizes the imaginarytime coordinate τ in small intervals Δτ, with the inverse of temperature T (projection time) being β = MΔτ. The interacting terms are transformed in singleparticle operators by means of a discrete HubbardStratonovich transformation, with the cost of including bosonic auxiliaryfields \({{\mathcal{S}}}_{{\bf{i}},\tau }\), in real space and imaginarytime coordinates, coupled to fermionic degrees of freedom. Monte Carlo methodologies are used for sampling \({{\mathcal{S}}}_{{\bf{i}},\tau }\). Throughout this article, we choose Δτt = 0.1, with β in unit of t. Finally, in view of the large autocorrelation time, a still unsolved challenge in the DQMC method for electronphonon systems, we restrict our analysis in the safe region of frequencies ω_{0}/t ≥ 1, when dealing with this approach (see also the Supplementary Notes 2). Detailed introduction for these methodologies can be found, e.g., in dos Santos^{65}, Gubernatis et al.^{66}, and Becca and Sorella^{67}.
Data availability
The datasets obtained during this work are available from the corresponding author upon reasonable request.
Code availability
Code is available upon reasonable request.
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Acknowledgements
We are grateful to Yuichi Otsuka for valuable discussions. Computational resources were provided by HOKUSAI supercomputer at RIKEN (Project ID: G19010), and CINECA supercomputer (PRACE2019204934). S.S. and N.C.C. acknowledge PRACE for awarding them access to Marconi at CINECA, Italy. N.C.C. also acknowledge the financial support from the Brazilian Agencies CAPES and CNPq. This work was partially supported by GrantinAid for Research Activity startup (No. JP19K23433) from MEXT, Japan.
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S.Y. and S.S. conceived the project. N.C.C implemented the determinant quantum Monte Carlo code, and K.S. the projective auxiliaryfield quantum Monte Carlo code. The simulations were carried out by N.C.C. and K.S. under the guidance of S.Y. and S.S. All authors participated to the discussions during the writing of the article.
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Costa, N.C., Seki, K., Yunoki, S. et al. Phase diagram of the twodimensional HubbardHolstein model. Commun Phys 3, 80 (2020). https://doi.org/10.1038/s4200502003422
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