Introduction

Strongly correlated materials, where the electronic structure cannot be approximated by the reductive band theory, have become a research frontier. In particular, two types of unconventional superconductivity have attracted considerable attention. One of them is the high-Tc superconductivity discovered in cuprates1. Although this class of materials has been investigated for nearly 40 years, the pairing mechanism remains an enigma2,3,4,5. The other type of unconventional superconductivity is the topological triplet-pairing superconductivity6,7,8, where electron fractionalizes into Majorana excitations9,10 and is the foundation for topological quantum computing11,12. Therefore, pursuing such exotic superconductivity in realistic compounds constitutes a stimulating research topic.

The single-band Hubbard model, as the prototypical model carrying the strong correlation effects, has been widely employed in the studies of many-electron systems13,14,15,16,17,18 as variants of this model are relevant to the two-dimensional (2D) cuprate superconductors. Besides, quasi-1D cuprate chains also constitute important class of strongly correlated materials that host intriguing correlated electron states and effects, e.g., the Tomonaga-Luttinger liquid (TLL) with spin-charge separation19,20,21. On the other hand, most theoretical studies of the ground-state and dynamical properties22,23,24,25 also lie in 1D as rigorous many-body simulations are more accessible using analytics26, exact diagonalization27,28,29,30, density matrix renormalization group (DMRG)31,32,33,34 and quantum Monte Carlo35,36,37,38,39,40,41. Since both the on-site interaction U and near-neighbor (NN) interaction V correspond to the electronic repulsion at different distances, previous numerical studies focused on the cases with repulsive U, V > 0 as supposed relevant to real materials36,41,42,43,44,45.

Most recently, a paradigm shift occurs as an in situ ARPES experiment on the 1D cuprate chain Ba2−xSrxCuO3+δ (BSCO) suggests an anomalously strong attraction V < 0 between NN electrons46. In contrast to the intrinsic electron-electron Coulomb repulsion, this attractive interaction is likely to be mediated by electron-phonon coupling47. Although a rigorous identification of its origin and precise quantification of its strength may require more experimental measurements, this discovery reveals the possibility of a positive-U and negative-V system in the transition metal oxides, where strong correlations and electron-phonon coupling widely coexist48,49,50,51,52,53. Such an effective attraction largely missed previously may serve as a key ingredient in both understanding the high-Tc superconductivity and enabling exotic quantum phases in correlated materials27,28,29,30,54,55,56,57,58,59,60,61,62,63. Therefore, an interesting question naturally arises: Does such an effective attraction V help establish superconducting pairing between the strongly correlated electrons?

To address this question, and also motivated by the recent experimental realization of such attractive-V extended Hubbard model (EHM, see Fig. 1a), we employ large-scale DMRG simulations and systematically explore its phase diagram. We especially focus on the possible realization of spin-triplet superconductivity while identifying all phases. At both half and quarter fillings, we have numerically determined the ground-state phase diagrams of the EHM, from which we identify a robust gapless TLL phase with a prominent spin-triplet superconducting (TS) pairing with algebraic singularity. In two dimensions, the triplet superconducting state is topologically non-trivial where the fractional excitation can emerge on the boundary10,64,65,66,67. However, quantum fluctuations are usually too strong in 1D such that interacting electrons in a Hubbard-type chain usually behave as a TLL, contradicting the mean-field and small-cluster predictions. Therefore in this paper, we refer to this emergent TLL phase with divergent superconducting susceptibility as a gapless TS phase.

Fig. 1: Extended Hubbard model and phase diagrams.
figure 1

a illustrates the BSCO compound and corresponding extended t-U-V Hubbard model with NN hopping t, on-site repulsive U, and NN attractive V terms. b, c show the quantum phase diagrams of the EHM at half and quarter fillings, respectively. There are triplet superconducting (TS), spin density wave (SDW), and Tomonaga-Luttinger liquid (TLL) phases. There are also three phase separation phases with singly (PS1) doubly (PS2) or fractionally (PSx) occupied clusters. The solid black lines represent the asymptotic phase boundary \(V=-\frac{U}{2}-\frac{8\ln 2}{3U}\) in the strong coupling limit, and the dashed line for V = −U/2 in the weak coupling limit30,54. The blue circle in c represents the parameters U = 8 and V = − 1 of the doped 1D cuprate chain BSCO46.

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Our main findings are summarized in Fig. 1. At half filling (see Fig. 1b), the TS phase survives only up to a finite Uc/t ≃ 2.3 and is absent when U > Uc. At quarter filling (see Fig. 1c), this TS phase extends to larger Us comparable to those in cuprates46,68,69. Between this TS phase and the regular phase separation (PS) phases with singly (PS1) and doubly (PS2) occupied clusters, we further identify an exotic PSx phase where the clustered electrons form the TLL and even TS states. With the model parameters determined from fitting dynamical data of BSCO46, our study reveals a close proximity of this doped cuprate chain to the p-wave superconductivity, and provide theoretical guide for realizing such gapless TS phase in 1D cuprate chains.

Results

EHM with NN attraction

We consider the minimal model for BSCO chain, i.e., EHM with on-site U > 0 and NN attraction V < 046, whose Hamiltonian reads

$$H=-t\mathop{\sum }\limits_{i=1,\sigma }^{L-1}({c}_{i\sigma }^{{{{\dagger}}} }{c}_{i+1\sigma }+{{{{{{{\rm{H.c.}}}}}}}})+U\mathop{\sum }\limits_{i=1}^{L}{n}_{i\uparrow }{n}_{i\downarrow }+V\mathop{\sum }\limits_{i=1}^{L-1}{n}_{i}{n}_{i+1},$$
(1)

where \({c}_{i\sigma }^{{{{\dagger}}} }\) (ciσ) is the electron creation (annihilation) operator, σ = ↑, ↓ labels the electron spin, and ni = ni↑ + ni↓ is the particle number operator at site i. Throughout the study, we set hopping amplitude t = 1 as the energy unit, and focus on the ground state phase diagrams at both half and quarter fillings. In this work, we employ DMRG method with non-Abelian symmetry implemented70,71 (see “Methods” and Supplementary Note 1).

To characterize various quantum phases, we compute the spin, charge, and pairing correlation functions. The spin-spin correlation is defined as F(r) = 〈Si ⋅ Sj〉, with Si(j) the spin operator at site i(j) and r ≡ j − i. The charge density correlation is defined as D(r) = 〈ninj〉 − 〈ni〉〈nj〉, where ni(j) is the particle number operator at site i(j). To characterize the superconducting (SC) pairing correlation, we consider both the spin-singlet (s-wave) pairing \({\Phi }_{{{{{{{{\rm{S}}}}}}}}}(r)=\langle {\Delta }_{{{{{{{{\rm{S}}}}}}}}}^{{{{\dagger}}} }(i){\Delta }_{{{{{{{{\rm{S}}}}}}}}}(j)\rangle\) with \({\Delta }_{{{{{{{{\rm{S}}}}}}}}}^{{{{\dagger}}} }(i)=\frac{1}{\sqrt{2}}({c}_{i,\uparrow }^{{{{\dagger}}} }{c}_{i+1,\downarrow }^{{{{\dagger}}} }-{c}_{i,\downarrow }^{{{{\dagger}}} }{c}_{i+1,\uparrow }^{{{{\dagger}}} })\), and the triplet (p-wave) pairing \({\Phi }_{{{{{{{{\rm{T}}}}}}}},s}(r)=\langle {\Delta }_{{{{{{{{\rm{T}}}}}}}},s}^{{{{\dagger}}} }(i){\Delta }_{{{{{{{{\rm{T}}}}}}}},s}(j)\rangle\) with three components \({\Delta }_{{{{{{{{\rm{T}}}}}}}},1}^{{{{\dagger}}} }(i)={c}_{i,\uparrow }^{{{{\dagger}}} }{c}_{i+1,\uparrow }^{{{{\dagger}}} }\), \({\Delta }_{{{{{{{{\rm{T}}}}}}}},0}^{{{{\dagger}}} }(i)=\frac{1}{\sqrt{2}}({c}_{i,\uparrow }^{{{{\dagger}}} }{c}_{i+1,\downarrow }^{{{{\dagger}}} }+{c}_{i,\downarrow }^{{{{\dagger}}} }{c}_{i+1,\uparrow }^{{{{\dagger}}} })\), and \({\Delta }_{{{{{{{{\rm{T}}}}}}}},-1}^{{{{\dagger}}} }(i)={c}_{i,\downarrow }^{{{{\dagger}}} }{c}_{i+1,\downarrow }^{{{{\dagger}}} }\) for s = 1, 0, − 1, respectively. Note that the EHM in Eq. (1) is SU(2) invariant so the above three components are degenerate in the spin-triplet channel, and we thus take the averaged \({\Phi }_{{{{{{{{\rm{T}}}}}}}}}(r)=\frac{1}{3}{\sum }_{s}{\Phi }_{{{{{{{{\rm{T}}}}}}}},s}(r)\) from our SU(2) DMRG calculations and compare it with ΦS.

Analytical results from the TLL theory

The TLL theory puts rigorous constraints72,73,74,75 on numerical results, which we always compare with and make use of in the analysis of our numerical data. In TLL, two-point correlation functions including the spin, charge and pairing correlations all decay in power law ~ r−α, with exponents α determined by two basic Luttinger parameters Kσ and Kρ, respectively related to the spin and charge degrees of freedom (see more details in the Supplementary Note 2). To accurately evaluate these intrinsic parameters, one can calculate the momentum-dependent spin structure factor Sm(k) and charge structure factor Sc(k), and then extract Kσ and Kρ.

For the current EHM in Eq. (1) with SU(2) spin symmetry, Kσ = 1 for the spin density wave (SDW), TLL, and TS phases with gapless spin excitations, while Kσ = 0 in the spin gapped phase PS2. Therefore, Kρ uniquely determines the power-law exponents α of various correlations: for charge and spin correlations there exist a uniform mode with exponent α0 = 2 and a 2kF mode with \({\alpha }_{2{k}_{{{{{{{{\rm{F}}}}}}}}}}=1+{K}_{\rho }\); for the pairing correlations ΦS and ΦT, they both have uniform modes with the same exponent αSC = 1 + 1/Kρ, which dominates over the spin and charge correlations when Kρ > 1. Consequently, the low-T behaviors of the staggered magnetic, charge, and pairing susceptibilities are also controlled by Kρ, i.e., \({\chi }_{{{{{{{{\rm{SDW}}}}}}}}} \sim {T}^{{K}_{\rho }-1}\), \({\chi }_{{{{{{{{\rm{CDW}}}}}}}}} \sim {T}^{{K}_{\rho }-1}\), and \({\chi }_{{{{{{{{\rm{SC}}}}}}}}} \sim {T}^{1/{K}_{\rho }-1}\). For Kρ > 1 or <1, these susceptibilities exhibit apparently distinct behaviors as T → 0. Thus, the Luttinger parameter constitutes an essential quantity characterizing the underlying phases of a 1D system. In practice, we extract the Luttinger parameter Kρ via a second-order polynomial fitting of Sc(k) in the small k regime41,45,73,76 (see Supplementary Note 3 for details). To minimize the boundary effect, we evaluate the correlation functions using sites away from both ends.

Quantum phase diagram at half filling

We summarize our main findings at half filling in the phase diagram of Fig. 1b, where the SDW, phase separation PS2 with doubly occupied sites clustered, and most remarkably, a TLL phase with prominent superconductive pairing is uncovered. To show the distinction of these phases, we present simulations along two typical paths in Fig. 2, namely, the U = 1.6 and U = 4 vertical cuts in the phase diagram.

Fig. 2: Results of EHM at half filling.
figure 2

a–c are with on-site repulsion of U = 1.6 and d–f of U = 4, with the corresponding mini phase diagrams also depicted. a, d show the results of the Luttinger parameter Kρ, (b, e) are the results of central charge c determined from the entanglement entropy scalings, and c, f are the spin structure factors at π. Here the Kρ results in a, d are obtained via the second-order polynomial fitting of Sc(k) in the small-k regime, with the error bars estimated by using different k-range for fitting (Supplementary Note 3). The insets in c, f show Sm(π) vs. L at various Vs. a–c reveal the system undergoes a transition from spin density wave (SDW) to triplet superconducting (TS) phase at Vc ≃ −1 and then enters the phase separation (PS2) regime for V < Vs ≃ −1.55. d–f show no intermediate phase but a direct first-order transition from SDW to PS2 at Vs ≃ −2.42.

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The Luttinger parameter Kρ clearly separates the U = 1.6 systems into three regimes. As the interaction strength increases to ∣V∣ > ∣Vc∣ ≃ 1 (but smaller than the phase separation transition strength ∣Vs∣, which will be discussed later), in Fig. 2a there exists an intermediate regime with Kρ > 1. We also compute the central charge c by fitting the entanglement entropy (see more details in Supplementary Note 4), and from Fig. 2b c is found to change from c ≃ 1 to about 2 for ∣Vc∣ < ∣V∣ < ∣Vs∣, confirming that the intermediate phase has both gapless spin and charge modes. On the other hand, also as shown in Fig. 2, for the U = 4 case Kρ remains small for all values of V and does not exceed 1 (see Fig. 2d) and the central charge remains c = 1 (Fig. 2e), showing the absence of such intermediate phase.

With further increase of the attractive interaction for either U = 1.6 or 4, the system eventually exhibits phase separation for ∣V∣ > ∣Vs∣. The critical strength Vs dependent on U is shown in Fig. 1 (see the detailed estimation of Vs in Supplementary Note 1). Specifically for the two selected cuts, we found Vs ≃ −1.55 for U = 1.6 (see Fig. 2a–c) and Vs ≃ − 2.42 for U = 4 (see Fig. 2d–f). In such a PS state, the clustered part consists of doubly-occupied sites and no singularity can be observed in various correlations. Therefore, we denote it as PS2 to distinguish from other PS phases discussed later.

Among these three phases in the U = 1.6 case (and for other interactions U < Uc ≃ 2.3, c.f., Fig. 1b), we are particularly interested in the intermediate one due to the signature of triplet pairing. As evidenced by the charge correlation results in Fig. 3a, the charge gap is closed by the attractive V term, and the Luttinger parameter Kρ can be fitted to be greater than 1 (see the inset of Fig. 3a, and more details in Supplementary Note 3). According to the TLL theory, the superconductive paring decays \({r}^{-{\alpha }_{{{{{{{{\rm{SC}}}}}}}}}}\) with the exponent αSC = 1 + 1/Kρ— smaller than the algebraic exponent (1 + Kρ) of both the charge and spin correlations when Kρ > 1— and thus constitutes the dominant correlation in the charge-2e channel, with an algebraically diverging pairing susceptibility χSC(T) for low temperature T.

Fig. 3: Correlation functions of the half-filled EHM at U = 1.6.
figure 3

Here we show a density-density, b spin-spin, c the spin-triplet pairing, and d singlet pairing correlations, with various attractive interactions − 1.5 ≤ V ≤ − 0.4. The color codes in the four panels are the same (following the representative color of each phase in Fig. 1b, as indicated by the legends in b, d). The dashed green line represents ∼ r−2 scaling that corresponds to Kρ = 1, separating the exponential and power-law behaviors of various correlations in a, c, d. The dotted dash line ∼ r−1 with exponent α = 1 in b represents long-distance scaling of the spin correlations in the SDW phase. The inset in a shows the charge structure factor at V = −1.5, whose second-order polynomial fitting (dashed cyan line) results in a Luttinger parameter of Kρ = 1.84.

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In the weak attraction regime ∣V∣ < ∣Vc∣, Kρ remains finite (< 1) in Fig. 2a due to the strong finite-size effects and a small charge gap for U = 1.6, which would converge to zero in the thermodynamic limit. Kσ = 1 due to the spin SU(2) symmetry. In Fig. 3b, an quasi-long range spin order with an algebraic exponent of αSDW = 1 appears, which has logarithmically diverging spin structure factor of Sm(k = π) (see Fig. 2c, f and the insets). This is well consistent with the SDW scenario with a finite charge gap and quasi-long range spin order (see Supplementary Note 5). On the other hand, for the intermediate phase in Fig. 2c Sm(π) ceases to increase vs. L, as the 2kF mode spin correlation decays faster than ~r−2 shown in Fig. 3b, which reveals a non-diverging magnetic susceptibility and thus rather distinct magnetic properties from that of the SDW phase.

Gapless triplet superconducting phase

As shown in Fig. 3c, d, it can be observed that both the singlet- (ΦS) and triplet-pairing (ΦT) exhibit power-law decay behaviors, and the latter with p-wave pairing symmetry clearly dominates over the former with the s-wave pairing symmetry. This is clearly demonstrated in Fig. 4a, where the strengths of the two correlations ΦT(r) and ΦS(r) are compared at a fixed distance r = 20. Though two pairing correlations are comparable in the SDW regime, ΦT(r) clearly surpasses ΦS(r) once entering the intermediate-V phase: the latter turns to decreasing, while ΦT(r) keeps increasing and becomes over one order of magnitude greater than ΦS(r).

Fig. 4: Dominance of the triplet over singlet pairing in the half-filled TS phase.
figure 4

Here we show the data of U = 1.6. a Singlet and triplet pairing correlations at a fixed (long) distance r = 20. The yellow strip represents the transition point Vc ≃ −1 from spin density wave (SDW) to triplet superconducting (TS) phase. b The calculated ratios ∣ΦT/ΦS∣, plotted in a log-log scale, exhibit excellent agreement with the scaling \({r}^{{K}_{\rho }-1}\) (blue solid lines).

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Such a dominance of the triplet pairing in the TS phase holds for different distances r other than the fixed distance r = 20 in Fig. 4a. This dominance is reflected in the spatial distribution of both pairing correlations in Fig. 3c, d. There we find ΦT firstly decays exponentially in the SDW phase (the blue dots), then exhibits power-law behaviors for ∣Vc∣ < ∣V∣ < ∣Vs∣ (the red dots), and decays again exponentially for ∣V∣ > ∣Vs∣ (the gray dots). We notice there is virtually no uniform but only 2kF mode in ΦS, as reflected in the smooth curves ΦS(r) × (−1)r−1 in Fig. 3d. For the gapless TS phase where we are most interested in, the dominance of ΦT is reflected by the comparison of Fig. 3c and d: ΦT(r) decays slower than r−2, while ΦS(r) decays faster than r−2 (Fig. 3d). More quantitatively, the ratio between these two pairing correlations ∣ΦT(r)/ΦS(r)∣ scales in power law \({r}^{{K}_{\rho }-1}\), since the leading scaling in ΦT and ΦS is \(1/{r}^{1+1/{K}_{\rho }}\) and \(1/{r}^{{K}_{\rho }+1/{K}_{\rho }}\), respectively (see Supplementary Note 2). We present such a power-law scaling extracted from our DMRG simulations in Fig 4b. Therefore, in the intermediate regime the pairing correlation ΦT dominates over ΦS not only in magnitude but actually in long-distance scaling, making it a rather unique gapless TS phase.

When compared to the phase diagram obtained in ref. 54, our DMRG results in Fig. 1b show some agreement on the existence of three phases, yet there are still noticeable differences. Particularly, our DMRG calculations identify the upper boundary of the TS phase in agreement with V = −U/2 obtained from the perturbation theory in the small U regime while it deviates from this line in the strong coupling regime. Consequently, in contrary to ref. 54 where the TS phase was shown extending to infinite U, our results in Fig. 1b suggest it can only survive up to Uc ≃ 2.3, located in a much narrower regime. On the other hand, when compared to more recent studies62,77 where the phase diagrams are only schematic, here we pinpoint the numerically accurate phase boundaries with large-scale DMRG calculations and reveal the predominant triplet quasi-long range TS pairing relevant to the realistic cuprate chain BSCO, decades after such a TS instability was proposed29,54.

Finite doping

Besides half filling, we have also explored the phases in the doped EHM systems. We first focus on the quarter filling, where the triplet pairing instability is approximately maximized, as will be discussed later. The extracted phase diagram is presented in Fig. 1c. Here, we select a cut along U = 4 and explain the properties of each phase in Fig. 5. Similar to half filling, the Luttinger parameter Kρ > 1 characterizes the intrinsic nature of the correlations and separates the U = 4 systems into four phases (see Fig. 5a). Particularly, for V < Vc ≃ −0.8, we identified a TS regime following the same principle as half filling, manifested as enhanced triplet and singlet pairing correlations. Between these two correlations, we evaluated their ratio ∣ΦT/ΦS∣ and found its envelop increasing monotonically as ∣V∣ enhances and exceeding 1 for V < Vc (see Fig. 5b), despite some oscillations with distance r. Note the two pairing correlations now show the same scaling at long distance. Importantly, the TS phase at quarter filling is significantly wider than that at half filling, particularly in the large U regime.

Fig. 5: Results of EHM at quarter filling.
figure 5

We show in a the mini phase diagram and Luttinger parameter Kρ, and b accordingly the ratio between triplet and singlet pairings ∣ΦT/ΦS∣ that becomes greater than 1 in the emergent TS phase. The Kρ results in the PSx and PS1 are computed within the clustered part (see Supplementary Note 3). c The charge distribution n(i) in TS (V = − 1), PSx (V = −1.7), PS1 (V = −2.4), and PS2 (V = −2.44) phases, respectively. d The charge densities n(L/2) measured at the center of the system are plotted versus V.

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Besides the TS regime, there are three different inhomogeneous PS phases, i.e., PS1, PSx, and PS2 in Fig. 1c, in the doped system. The real-space charge distributions n(i) are shown in Fig. 5c, from which we see that in the PS phases the electrons cluster with filling n = 1, 2 or x ∈ (1/2, 1]). To track the evolution among these PS phases when V changes, we pick the center of the system as a representative, which always lies in the filled domain in a PS state due to the open boundary, and extract n(i = L/2) for different U and V strengths in Fig. 5d. This filling density starts with n(L/2) = 0.5 (i.e., the TLL and TS phases) and deviates from the uniform quarter filling when ∣V∣ is stronger than certain transition value. As n(L/2) = x is not a fixed integer value but varies between 0.5 and 1, we denote this regime as PSx. For small U, like U = 2, the system jumps from PSx to PS2 at a second transition point. In contrast, this transition is preceded by a third PS phase for large U > Uc ≃ 2.3 (the same as that of half filling). Taking U = 4 as an example, PSx firstly transits into an n(L/2) = 1 phase (denoted as PS1), and then jumps into PS2 as ∣V∣ further increases. For the doped cases with filling factors other than 1/4, the quantum phase diagram is qualitatively similar to that of Fig. 1c. The phase boundaries of PS1 and PS2 actually remain intact for other doping since they reflect the local energy relation between singly and doubly occupied states. The quantum many-body states in the clustered part of the three PS phases — PS1, PS2, and PSx — only depends on the interaction parameters U and V.

The existence of the PSx phase was missed in early studies on the same model29,54, and the distinct feature of PSx is the clustered electrons that constitute a TLL liquid with fractional filling. With x continuous tuned by V, the clustered part of PSx can also become close to half filling in terms of density, i.e., x = 1. Nevertheless, it is distinct from that in the PS1 phase, as the clustered electrons in the latter form a charge gapped SDW instead of a gapless TLL. Even more interestingly, we can also identify a Kρ > 1 regime with significant TS pairing correlations in the clustered part of PSx, showing the existence of gapless TS cluster in (at least part of) the PSx phase (see more details in Supplementary Note 6).

Discussion

Our simulation is motivated by but not restricted to the recently extracted attractive-V extended Hubbard model for 1D cuprate BSCO from experiments46. Previously, there were weak-coupling renormalization group (RG) and functional RG studies of the half-filled EHM that suggested the presence of TS pairing based on the field-theoretical analysis 54,55,61,63. Besides, there were also ED and DMRG calculations of the phase diagram for half-filled or doped cases28,29,30,56,57,59,60,62. The TS phase is characterized by dominant triplet superconducting pairing correlation, whose long-distance scaling analysis, however, was missing in previous studies.

In this work, we employ DMRG—the method of choice for 1D correlated systems—to investigate the EHM with both on-site repulsive and near-neighbor attractive interactions. At both half and quarter fillings, we identify a prominent gapless TS phase with the p-wave pairing induced by the attractive interactions. Different from the long-range order (hidden) assumption in the context of mean-field theory, the p-wave superconducting order identified in this correlated 1D chain is quasi-long-ranged: the triplet pairing correlation ΦT decays as a power-law at long distance and presents as the dominant charge-2e excitations in the gapless TLL, and specially, at half filling it dominates over the singlet pairing ΦS also in large distance scaling. Such dominance results in divergent triplet superconductive susceptibility at low temperature. This phenomenon can be detected by the spectral depletion in ARPES or the Drude peak in optical conductivity, both of which are accessible for in situ synthesized quasi-1D materials.

Besides the qualitative identification of various phases, our work also pushed the exact phase diagram to a quantitative level. This is important because the interest in the 1D EHM (with V < 0 and U > 0) is no longer restricted to theoretical discussions. It has been proposed as the underlying model for 1D cuprate chains46,47. Therefore, a quantitatively accurate phase diagram, especially with doping dependence, will be essential for up-coming experimental investigations of the TS phase in the 1D cuprate BSCO. For example, it is noted that the phase boundaries in Fig. 1, i.e., the critical strengths of V for entering the TS phase, are U dependent, and they can be determined analytically at quarter filling in the U → ∞ limit78,79. In this limit, the Luttinger parameter \({K}_{\rho }=1/[2+(4/\pi )\arcsin (v)]\) with v = V/2, which exceeds 1 when \(| V| \ge \sqrt{2}\). According to ref. 46 the effective model parameters for 1D cuprate chain BSCO were proposed as U ≃ 8 and V ≃ −1, where the effective attraction V is still slightly below this threshold.

To search for TS in larger parameter space, we further explore the full doping dependence. To approximate the realistic materials, we fix U = 8 and three different values of V, and evaluate the Luttinger parameter Kρ for a wide range of doping. As shown in Fig. 6, the uniform TS phase characterized as Kρ > 1 can be realized only if ∣V∣ > 1.2 (and ∣V∣ ≲ 1.7 before PSx sets in), in order to exhibit prominent superconducting instability below 40% doping, the maximal accessible doping at current experimental conditions. Therefore, the doped BSCO resides on the boundary to a TS phase (as also indicated in Fig. 1c), and a slight reduction of on-site U or enhancement of near-neighbor attraction V may drive it into the TS phase—both can be achieved by manipulating the electron-phonon coupling either inside the crystal or via a substrate47.

Fig. 6: Prediction for possible triplet superconducting phase in 1D cuprate chain.
figure 6

We show the Luttinger parameter Kρ versus doping δ for different attractive interactions V. In the highlighted regimes with Kρ > 1, there exists superconductivity phase for sufficiently strong attraction V and large doping δ. The blue circle represents the cuprate Ba2−xSrxCuO3+δ with the parameters V = − 146 and δ ≃ 40%, and a slight enhancement of the attraction can drive the compound to the triplet superconductive phase.

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Moreover, the minimal EHM model proposed in ref. 46 can well explain the experimental observations in the BSCO chain, yet the obtained parameters can still have ~20–30% error bars, i.e., the U and V may extend a range. Recently, a numerical study actually suggests a smaller U = 5.0t in BSCO chain69, which places the compound in the TS phase according to our phase diagram (Fig. 1b) at quarter filling.

Given that, the TS phase identified in our simulations may motivate further investigation and manipulation of cuprates towards a p-wave topological superconductor. The couplings between the cuprate chains may open a charge gap and introduce edge modes that can be very useful in future quantum technologies. Due to the chemical and structural similarity between 1D and 2D cuprates, our results of the TS phase in the attractive EHM here shed light on and call for further many-body studies of the superconductivity in the EHM of higher dimensions80,81,82,83.

Lastly, our conclusion is not restricted to the cuprate BSCO. Considering the widely existing electron repulsion and electron-phonon coupling48,49,50,51,52,53, this model with a repulsive U and an attractive V may also be applicable, as a low-energy approximation, for other transition-metal oxides. These different cuprate compounds and other materials may exhibit different microscopic parameters (U and V) due to their distinct chemical environments, and the rich quantum phases revealed in the EHM model studies here may find their interesting materialization in these systems.

Methods

Density matrix renormalization group

We perform DMRG calculations with the charge U(1) and spin SU(2) symmetries implemented through the tensor library QSpace70,71, and compute system sizes up to L = 512 to obtain the spin, charge, and superconductive correlations, etc, with high precision. In the calculations, we retain up to m* = 2048 multiplets, equivalent to m ≈ 5000 U(1) states, which render small truncation errors ϵ ≲ 10−7. We use the open boundary conditions as in conventional DMRG calculations. Due to the existence of attraction V, particularly near the PS phase one needs to introduce pinning fields at both ends and perform sufficient numbers of sweeps (even over 100 times) to fully converge the results, e.g., the charge distribution along the chain (see Supplementary Note 1).