The Influence of Extended Interactions on Spin Dynamics in One-dimensional Cuprates

The origin of high-temperature superconductivity that has been found in layered, quasi-two-dimensional (2D) cuprates remains elusive despite concerted investigations over the last few decades. From the perspective of numerical simulations, although simplified Hamiltonians, such as the Hubbard model and related variants, have produced rich physics, seemingly relevant to the cuprates [1, 2, 3], insufficient evidence exists for the presence of $d$-wave superconductivity in the ground state. Quasi-long-range superconductivity has been reported only on small width cylinders with a strong competition from coexisting charge order [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Moreover, it seems superconducting correlations decay exponentially for wider clusters, indicating that superconductivity may be absent for parameters thought to be relevant to the hole-doped cuprate compounds.

While numerical difficulties in reaching the 2D limit may prevent us from saying anything definitive about superconductivity in these simplified models, additional ingredients may well be needed to provide the crucial boost for superconductivity. Assessing the impact of missing ingredients, such as phonons, whose influence has manifest as kinks or replica bands in photoemission measurements [14, 15, 16, 17, 18], or long-range interactions, is even more challenging in 2D modeling due to the additional numerical difficulty associated with adding more degrees of freedom and the associated expansion of the effective Hilbert space, and the growth of long-range entanglement. Our task may be made easier, with more numerical control and better theoretical understanding, by turning to the simpler, yet structurally similar, quasi-one-dimensional (1D) cuprates.

Recent synthesis progress on the doped 1D-chain cuprate $\mathrm{Ba}_{2-x}\mathrm{Sr}_{x}\mathrm{CuO}_{3+\delta}$ [19] provides an excellent opportunity for testing theoretical models against experiments. Angle-resolved photoemission spectroscopy (ARPES) data on this doped 1D-chain in conjunction with numerical simulations revealed the presence of a strong, extended, attractive interaction [19]. Such an effective strong and extended attractive interaction likely originates from extended electron-phonon (el-ph) couplings [20]. It was found in later numerical work [21], that the extended el-ph coupling reproduces well the doping dependence of salient ARPES spectral features, which has been shown to enhance superconducting pairing correlations in 1D and have important implications for realizing a $d$-wave superconducting ground state in the structurally similar layered 2D cuprates. This extended el-ph coupling can significantly suppress the static spin-spin correlation [21] otherwise obtained from the Hubbard model, while only slightly affecting the exponent of the single particle correlations. This motivates us to unravel the spin-spin correlation in frequency through the dynamical spin structure factor $S(q,\omega)$, which can be used to further refine how the el-ph interactions impact the spin dynamics beyond the simple Hubbard model. As we will show, $S(q,\omega)$ may provide a sensitive probe on the presence and influence of extended el-ph interactions in the 1D cuprates.

In this work, we compute the dynamical spin structure factor $S(q,\omega)$ influenced by extended, attractive couplings on a 1D chain for various hole doping concentrations $\delta$. We demonstrate that the addition of such extended interactions can introduce significant qualitative and quantitative changes to $S(q,\omega)$ as a function of $\delta$. The two-spinon excitations at the zone edge significantly broaden and soften, even becoming gapless across the momentum range $(1-\delta)\pi\leq q\leq\pi$ when the extended interactions come from el-ph coupling, while also leading to an apparent shift in the “2$k_{\textrm{F}}$” momentum position towards $\pi$, compared to the simple Hubbard model [$2k_{\textrm{F}}=(1-\delta)\pi$] .

The Hubbard model (HM) is purely electronic

where $\hat{c}^{\dagger}_{i\sigma}$ ($\hat{c}_{i\sigma}$) is the charge creation (annihilation) operator on site $i$ for spin $\sigma$, $\hat{n}_{i\sigma}$ is the charge number operator on site $i$ for spin $\sigma$, and $U$ is the on-site repulsion. To avoid confusion with the time variable $t$, we use $t_{h}$ to denote the hopping integral. We consider two forms of extended, attractive couplings to modify this simple HM. In one case, we introduce an effective nearest-neighbor attractive interaction, which has been used in prior work to well-reproduce the apparent single-particle spectra from ARPES experiments [19], to produce an extended-Hubbard model (HM+$V$) given by

where $\hat{n}_{i}$ and $\hat{n}_{j}$ are total charge number operators on neighboring sites. In the other case, we add local optical phonons with coupling to the on-site and nearest-neighbor charge densities to the HM, resulting in a Hubbard-extended Holstein model (HM+$g_{0}$+$g_{1}$) [20, 21]

where $\hat{a}^{\dagger}_{i}$ and $\hat{a}_{i}$ are the phonon ladder operators on site $i$, $\hat{n}_{i}$ is the total charge number operator on site $i$, $\omega_{0}$ is the phonon frequency, $g_{0}$ is the on-site el-ph coupling, $g_{1}$ is the nearest-neighbor el-ph coupling, and $\left<ij\right>$ sums over nearest-neighbors.

To solve for the ground states of these models, we use the density matrix renormalization group (DMRG) [22, 23] method; and for the HM+$g_{0}$+$g_{1}$ we employ a local basis optimization (LBO) [24] for the phonon degrees of freedom in Eq. 3. We then use time-dependent DMRG (tDMRG) [25, 26, 27], with a dynamical LBO [28] for the HM+$g_{0}$+$g_{1}$, to calculate the correlator

which is measured at uniformly spaced discrete time steps $t_{n}=n\delta t$, $n=0,1,\cdots,N$, until reaching the maximum time $T=N\delta t$. $S(q,\omega)$ is then obtained by Fourier transform. To reduce the effect of the cutoff at finite time $T$, a decaying window function $\mathcal{W}_{\sigma}(t)$, where $\mathcal{W}_{\sigma}(T)\approx 0$, is multiplied to the time signal. The resulting spectra is

We use a window function $\mathcal{W}_{\sigma}(t)=e^{-\frac{\sigma^{2}}{2}t^{2}}$, whose Fourier transformation $\mathcal{F}\left[\mathcal{W}_{\sigma}(t)\right]$ is a Gaussian with standard deviation $\sigma$. This window function broadens the spectra

We compute and compare $S(q,w)$ at various doping levels $\delta$ for the three models: HM, HM+$V$ and HM+$g_{0}$+$g_{1}$. Unless otherwise stated, the spectra have been evaluated for parameters $U=8t_{h}$ and $V=-t_{h}$ for the HM+$V$ or $\omega_{0}=0.2t_{h}$, $g_{0}=0.3t_{h}$, and $g_{1}=0.15t_{h}$ for the HM+$g_{0}$+$g_{1}$. These parameters have been shown to well-reproduce features in the experimental single-particle spectra as measured by ARPES [19, 21].

We first show results for relatively long chains with $L=80$ and relatively long total times to produce high resolution spectra with small finite-size effects to elucidate the spectral changes for representative doping $\delta=20\%$. As shown in Figs. 1(a) and (b), both the simple HM and HM+$V$ have gapless spectra at momentum $q_{0}=(1-\delta)\pi$, and with increasing momentum from $q_{0}$ to $\pi$, the excitation gap grows monotonically. Similar to the overall renormalization of the single-particle bandwidth and Fermi velocity with the introduction of the extended Hubbard interaction [19, 21], the spectra for the HM+$V$ [shown in panel (b)] between $q_{0}$ to $\pi$ are noticeably softer than the spectra of the simple HM. In other words, a spectral gap at $q=\pi$ extends to higher energies in the absence of extended interaction $V$.

The introduction of extended el-ph coupling produces qualitative differences in $S(q,\omega)$ compared to both the HM and HM+$V$. For the HM+$g_{0}$+$g_{1}$, shown in Fig. 1(c), the lowest excitation peak becomes significantly broadened across momenta $q\in(q_{0},\pi)$, with spectral weight extending to very low energy (approaching $\omega=0$). As a comparison of the three models, we plot the energy distribution curves (EDC) at $q=\pi$ in Fig. 1(d), highlighting both the softening of the gap in the HM+$V$ and the significant broadening and low energy spectral weight caused by the extended el-ph couplings.

An additional, but not so obvious, difference induced by the extended el-ph coupling occurs in the apparent “$2k_{F}$” position. We show the momentum distribution curves (MDC) at $\omega=0$ in Fig. 1(e). For both the HM and HM+$V$, the spectral peak position is well defined at $q_{0}=(1-\delta)\pi$. The introduction of extended el-ph coupling broadens and shifts the peak towards $\pi$.

Due to the numerical complexities associated with the addition of phonons, we perform the remaining simulations on short 40-site chains. The resulting spectral resolution is sufficient to demonstrate the impact of extended interactions on $S(q,\omega)$ as a function of hole doping $\delta$, and the differences between an effective attractive interaction $V$ and extended el-ph coupling.

In Fig. 2, we show $S(q,\omega)$ for different models (rows) with increasing doping $\delta$ (columns). The time-dependent spin-spin correlators in real-space $S(r,t)$ can be found in Fig. S2 of the Supplementary Material. Even with the lower resolution afforded on the 40-site chain, one still sees that both the HM and HM+$V$ at doping level $\delta$ possess gapless $S(q,\omega)$ spectra at momentum $q_{0}=(1-\delta)\pi$. In each case, for momenta from $q_{0}$ to $\pi$, this excitation gap grows monotonically. As doping $\delta$ increases, $q_{0}$ becomes smaller and the gap at $\pi$ becomes larger (see Fig. 2 rows A and B). Compared to the simple HM, the HM+$V$ has a smaller excitation gap at $q=\pi$ across all doping levels, as also seen in the higher resolution spectra of Fig. 1.

For the HM+$g_{0}$+$g_{1}$ with $\delta>0$ (Fig. 2 row C), the lowest excitation peak is significantly broadened across momentum $q\in(q_{0},\pi)$ at all doping levels, and there is noticeable spectral weight extending to very low energy (even approaching $\omega=0$ at low doping).

As a comparison of the three models, we plot the energy distribution curves (EDC) at $q=\pi$ in the right column of Fig. 3, demonstrating broadened peaks and low energy spectral weight caused by the extended el-ph couplings. In terms of the peak positions at $q=\pi$, compared to the HM, the HM+$g_{0}$+$g_{1}$ has lower energy peaks at low doping ($\delta<30\%$), but the peak positions at higher doping are more comparable; however, at all doping levels the spectra of the HM+$g_{0}$+$g_{1}$ are significantly broadened compared to both the HM and HM+$V$. We also show the momentum distribution curves (MDC) at $\omega=0$ obtained from Fig. 2 in the left column of Fig. 3. For both the HM and HM+$V$, the spectral peaks are well defined at $q_{0}=(1-\delta)\pi$. With extended el-ph coupling, the peaks broaden and shift toward $\pi$ at small doping, as observed with higher resolution on the 80-site chain.

We have shown that the extended el-ph coupling induces significant changes to the spin dynamics across a wide range of doping levels. Specifically, without the extended el-ph coupling, $S(q,\omega)$ is gapless at momentum $q_{0}=2k_{F}=(1-\delta)\pi$ and has a gap that widens from $q_{0}$ to $\pi$. The extended el-ph coupling significantly broadens the excitations across this momentum range and introduces noticeable spectral weight down to zero energy for doping levels up to at least 50%, as examined here. Compared to HM, the peak positions at $q=\pi$ are softened at smaller doping but remain comparable, yet broad, at higher doping levels. There is also a shift of $q_{0}$ from $2k_{F}$ towards $\pi$ caused by the extended el-ph coupling. We also have demonstrated that an effective attractive interaction $V$ can not emulate well all these quantitative, or even qualitative changes in $S(q,\omega)$ induced by the extended el-ph coupling.

Compared to the enhancement of the holon folding branch seen in the single particle spectral function $A(q,\omega)$, which is relatively weak and disappears quickly as the doping level increases, [19, 21] the impact of el-ph coupling can be more readily observed in $S(q,\omega)$ across a wide range of doping due to the modification of spectral weight. The position of $q_{0}$, the peak positions at $q=\pi$, and the general shape of both MDC and EDC cuts at lower doping can provide vital clues about the strength of extended el-ph coupling in these materials. Our work points out that a promising and perhaps more sensitive experimental assessment of extended el-ph coupling in the doped 1D cuprates can come, for example, from measurements of $S(q,\omega)$ by resonant inelastic X-ray scattering (RIXS) [29].