Collective spin and charge excitations in the $t$-$J$-$U$ model of high-$T_c$ cuprates

The $t$-$J$-$U$ model of high-$T_c$ copper-oxide superconductors incorporates both the on-site Coulomb repulsion and kinetic exchange interaction and yields a semi-quantitative description of the static properties of those materials. We extend this analysis to dynamic quantities and address collective spin- and charge excitations in the correlated metallic state of the $t$-$J$-$U$ model. We employ VWF+$1/\mathcal{N}_f$ approach that combines the variational wave function (VWF) approach with the expansion in the inverse number of fermionic flavors ($1/\mathcal{N}_f$). It is shown that the resonant (paramagnon) contribution to the dynamic magnetic susceptibility remains robust as one interpolates between the Hubbard- and $t$-$J$-model limits, whereas the incoherent continuum undergoes substantial renormalization. Energy of the collective charge mode diminishes as the strong-coupling limit is approached. We also introduce the concept of effective kinetic exchange interaction that allows for a unified interpretation of magnetic dynamics in the Hubbard, $t$-$J$, and $t$-$J$-$U$ models. The results are discussed in the context of recent resonant inelastic $x$-ray scattering experiments for the high-$T_c$ cuprates.


I. INTRODUCTION
High-temperature (high-T c ) copper-oxide superconductors (SC) serve as paradigmatic strongly-correlated systems, evolving from the antiferromagnetic (AF) Mott insulating state, through high-T c SC state, to normal metal phase as a function of chemical doping. The common theoretical frameworks, used to describe them, are based on (extended) Hubbard and t-J-models, yet a unified quantitative description of static-and single-particle properties of high-T c cuprates for a fixed set of microscopic parameters has not been achieved so far within those schemes. In effect, extended t-J-U Hamiltonian, encompassing both Hubbard and t-J models as special cases, has been proposed and demonstrated to yield a consistent description of principal experimental data for high-T c cuprates within variational scheme. [1][2][3][4] However, the t-J-U -model studies have been hitherto restricted to static properties, wheres recent developments in spectroscopic techniques (in particular, resonant inelastic x-ray scattering (RIXS)) provide comprehensive evidence for the relevance of collective excitations across the phase diagram of high-T c cuprates.  In this respect, the t-J-U model remains largely unexplored and is yet to be tested as a tool to study many-particle dynamics.
Here we analyze the structure of dynamical spin and charge susceptibilities in the paramagnetic metallic state of the two-dimensional t-J-U model and relate its predictions to those of the Hubbard and t-J models that are used extensively [27][28][29][30][31][32][33] to study collective modes in correlated electron systems. We employ VWF+1/N f scheme, [28][29][30] combining Variational Wave Function (VWF) approach in its diagrammatic form with the field-theoretical 1/N f expansion (N f denotes the number of fermionic flavors). The latter has been recently benchmarked 29 against determinant quantum Monte-Carlo for small systems. This allows us to incorporate both the effect of local correlations and longrange collective excitations, required to be able to analyze the strong-coupling (t-J-model) limit. We find that the resonant part of the magnetic response (paramagnon mode) remains comparable within Hubbard, t-J, and t-J-U models, provided that the parameters of those three Hamiltonians are appropriately mapped onto each other. On the other hand, the incoherent part of the spectrum is substantially affected by variation of the Hubbard U . We also find that the charge mode energy undergoes reduction as one moves from the Hubbard to the t-J-model regime, reflecting enhanced band renormalization effects near the t-J-model limit. This analysis supports the t-J-U model as a tool to study semi-quantitatively collective modes in high-T c cuprates.

II. MODEL AND METHOD
We employ the t-J-U model on two-dimensional 200 × 200 square lattice, given by the Hamiltonian We retain only nearest-and next-nearest hopping integrals, t ≡ −0.35 eV and t ≡ 0.25|t|, respectively. The on-site Coulomb repulsion, U , and exchange coupling, J, are treated as free parameters. To interpret the results on the U -J plane, we introduce effective kinetic exchange interaction, J eff ≡ J + 4t 2 U , combining J with the secondorder kinetic exchange that results from the canonical perturbation expansion 34 for the Hubbard model. Unless stated otherwise, we restrict the parameter space by setting J eff ≡ 0.2 eV so that the exchange coupling is determined by U ≥ 7|t| as J = J eff − 4t 2 U . In the Hubbard model limit (J = 0, U = 7 |t|), this choice yields a semiquantitative agreement with measured paramagnon spectra for the lanthanum cuprates. 28 Moreover, the selection of sizable J eff favors AF correlations and thus places the arXiv:2107.02152v1 [cond-mat.str-el] 5 Jul 2021 system away from ferromagnetism [35][36][37][38] (cf. the phase stability analysis below). Hereafter we set the hole concentration to δ = 0.16 and temperature k B T = 0.4|t| to ensure stability of the paramagnetic state against fluctuations.
The model is solved using VWF+1/N f approach in the local-diagrammatic (LD) variant, LD f + 1/N f . 29 First, the energy functional E var ≡ Ψ var |Ĥ|Ψ var is constructed based on the t-J-U Hamiltonian (1) and the trial state, |Ψ var ≡ C iP i |Ψ 0 , where |Ψ 0 represents Slater determinant to be determined self-consistently, introduces local correlations into |Ψ var , and C is the normalization factor. The correlators,P i , are expressed in terms of the local many-particle basis on site i, encompassing empty (|0 i ), singly-occupied (|↑ i , |↓ i ), and doubly occupied (|↑↓ i ) configurations. The six coefficients, λ 0 , λ σσ (σ, σ =↑, ↓), and λ d , serve as variational parameters. By application Wick's theorem, energy is then expressed as E var = E var (P, λ), where P is a vector composed of two-point correlation functions (lines) in the form P iσ,jσ = Ψ 0 |ĉ † iσĉ jσ |Ψ 0 , and λ denotes correlator parameters. We evaluate E var within specialized diagrammatic scheme 39,40 and retain only local diagrams, which results in the so-called local diagrammatic (LD) approximation. 29 The latter incorporates multiple-loop diagrams and goes beyond the renormalized mean-field theory already at the zeroth-order (saddle-point) level. Both P and λ are then treated as (imaginary-time) dy-namical fields, and their quantum and classical fluctuations around this correlated state are studied at the leading order in 1/N f expansion (note that the fields are subjected to additional constraints, cf. Ref. 29). This allows us to calculate imaginary-time dynamical spin and charge susceptibilities (χ s and χ c , respectively), which are analytically continued to real frequencies as iω n → ω + i0.02|t|.

III. RESULTS
In Fig. 1, the calculated imaginary parts of spin (top) and charge (bottom) dynamical susceptibilities for the t-J-U model (1)  . As is apparent from Fig. 1(a)-(e), the magnetic spectrum separates into intense and dispersive paramagnon contribution (ranging up to ∼ 0.3 eV) and the incoherent part. The latter may be interpreted as particle-hole continuum, which has been verified explicitly in the Hubbard-model limit by comparison with correlated Lindhard susceptibility. 30 Remarkably, welldefined paramagnon persists along the anti-nodal (Γ-X) line and the magnetic Brillouin zone boundary (X-M/2) in the metallic state (δ = 0.16), but it is absent in the nodal (Γ-M ) direction. This shows that robust magnetic excitations may arise in strongly corre- lated systems and resist overdamping, even without longrange magnetic order. In the Hubbard-model limit, those highly anisotropic paramagnon characteristics have been recently shown to semi-quantitatively reproduce experimental inelastic neutron scattering and RIXS data for selected high-T c superconductors. 28,30 As the main finding of the present study, we demonstrate weak dependence of the peak paramagnon energy on U for a fixed value of J eff [cf. Fig. 1(a)-(e)]. This supports the proposed here interpretation of J eff as the scale controlling paramagnons in the metallic state of the t-J-U model close to AF instability and should allow us to reconcile the Hubbard-model results for dynamic quantities with the t-J-U -(V ) model analysis 1-4 of equilibrium properties. The particle-hole excitations exhibit qualitatively different behavior and are shifted to lower energies as the on-site Coulomb interaction increases, which can be attributed to correlation-induced narrowing of singleparticle bandwidth. The coherent-and incoherent magnetic excitations are thus, to large extent, decoupled and governed by distinct energy scales (J eff and renormalized bandwidth, respectively).
Charge excitations may be decomposed in an analogous manner into coherent-and incoherent components, with a well-defined mode emerging above the contin- uum. From Fig. 1(f)-(j) it follows that the charge mode energy undergoes reduction as the system evolves from the Hubbard-to the t-J-model limits. Remarkably, the charge-mode energy approaches zero near the Γ point as a consequence of neglecting long-range threedimensional Coulomb repulsion in the model (1), whose singular small-k behavior is known to open the Γpoint plasmon gap and induce dimensional crossovers in layered systems. 41 Both effects have been observed experimentally, 22,24,26 so long-range interactions are necessary to reliably describe plasmons in the cuprates. In Fig. 2 we display the energy dependence of the calculated imaginary parts of spin (left panels) and charge (right panels) dynamical susceptibilities for the Hubbard model (U = 7; red lines), t-J-U model (U = 16; blue lines), and the t-J model (U = ∞; green lines), all with J eff ≡ 0.2 eV. The panels correspond to representative points in the Brillouin zone, detailed inside the plot (wave vectors are related to h and k as k = ( 2π a h, 2π a k), with a being lattice constant). Several features of the spectra may be noted. First, a well-defined peak emerges from the continuum only along the anti-nodal (Γ-X) line, with maximum intensity corresponding to energy ∼ 0.  energies for increasing U , with an intensity peak systematically building up at its boundary. Remarkably, this second peak appears only for U substantially exceeding the bare quasiparticle bandwidth [cf. red (U = 16|t|) and green (U = ∞) lines in Fig. 2]. The lack of a double-peak signature in RIXS experiments supports thus the t-J-U over the t-J model as an appropriate starting point for variational studies of collective dynamics in the cuprates.
For completeness, in Fig. 3 the stability of paramagnetic state against spin (a) and charge (b) fluctuations is verified for the parameter range encompassing that used in the analysis of collective excitations (cf. Figs. 1 and 2). The static susceptibilities are displayed along the highsymmetry M/2-X-Γ-M contour (all curves correspond to J eff = 0.2 eV and varying values of U and J, detailed inside the panels). Both magnetic and charge responses remain finite in the entire parameter range, implying local stability of the paramagnetic state against fluctuations. The spin susceptibility attains a maximum at the M point, reflecting the dominant role of AF correlations. The charge susceptibility is peaked at the Γ point, which is characteristic of the model with short-range interactions. The singular tail of the Coulomb repulsion tends to suppress the charge response around the Γ-point. 30,42 Finally, we examine the dependence of the paramagnon energy on J eff . In Fig. 4 Fig. 4 shows that the paramagnon energy is indeed sensitive to J eff [(a)-(b)], whereas renormalization of charge mode energy due to variation of J eff is less significant [(c)-(d)]. This concludes the analysis of J eff as the energy scale governing paramagnon dynamics in the t-J-U model with dominant AF correlations. The opposite is true for the charge excitations, which is sensitive to the value of U .

IV. SUMMARY AND OUTLOOK
We have analyzed the structure of spin and charge excitations in the t-J-U model (1) of high-T c cuprates within the VWF+1/N f scheme. The effective exchange interaction, J eff ≡ J + 4t 2 U , has been identified as the energy scale controlling the anti-nodal paramagnon energies close to AF ordering, which rationalizes experimentally reported similarity of Γ-X magnetic excitations in metallic and AF insulating state. It also means that the paramagnon dynamics is governed predominantly by shortrange spin-flip correlations (either introduced through J or those originating from second-order exchange ∝ t 2 /U ), the same that provide pairing potential within the localpairing scenario of high-T c SC. Whereas generic occurrence of those magnetic excitations in the cuprates has revived the discussion about their role in high-T c SC, 18,43 our analysis supports the view that SC and persistent paramagnons may share a common microscopic origin, but need not to strictly couple to each other. The latter statement is consistent with the observation of highenergy paramagnons also in the overdoped regime, where SC is already suppressed. 5, 19 We have also noted that the incoherent magnetic excitations are governed predominantly by renormalized single-quasiparticle bandwidth and thus remain sensitive to the magnitude of the on-site Coulomb repulsion. For U much larger than the bare bandwidth, a sharp feature builds up at the threshold of a particle-hole continuum, which is not unambiguously observed in experiments. This circumstance favors the t-J-U model with intermediate U over the t-J model as a starting point for the discussion of the paramagnons in hole-doped high-T c cuprates. Similar conclusions follow from former studies of equilibrium and single-particle properties. [1][2][3][4] We have also demonstrated a substantial reduction of charge-mode energy with increasing U , which provides the flexibility required to interpret experimental data in real materials. In particular, the above characteristics of collective excitations in the t-J-U model may allow us to reconcile the observed persistent anti-nodal paramagnons with particularly low-energy acoustic plasmons 26 in lanthanum cuprates. Finally, the role of J eff is expected to be diminished in the regime of extremely large U and small J, where ferromagnetic correlations may dominate. [35][36][37][38] Those aspects should be the subject of a separate study.
A separate future question concerns the influence of the collective excitations on the pairing induced by the correlations.