Abstract
In quantum materials, the electronic interaction and the electronphonon coupling are, in general, two essential ingredients, the combined impact of which may drive exotic phases. Recently, an anomalously strong electronelectron attraction, likely mediated by phonons, has been proposed in onedimensional copperoxide chain Ba_{2−x}Sr_{x}CuO_{3+δ}. Yet, it is unclear how this strong nearneighbor attraction V influences the superconductivity pairing in the system. Here we perform accurate manybody calculations to study the extended Hubbard model with onsite Coulomb repulsion U > 0 and nearneighbor attraction V < 0 that could well describe the cuprate chain and likely other similar transitionmetal materials with both strong correlations and lattice effects. We find a rich quantum phase diagram containing an intriguing TomonagaLuttinger liquid phase — besides the spin density wave and various phase separation phases — that can host dominant spintriplet pairing correlations and divergent superconductive susceptibility. Upon doping, the spintriplet superconducting regime can be further broadened, offering a feasible mechanism to realize pwave superconductivity in realistic cuprate chains.
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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 highT_{c} superconductivity discovered in cuprates^{1}. Although this class of materials has been investigated for nearly 40 years, the pairing mechanism remains an enigma^{2,3,4,5}. The other type of unconventional superconductivity is the topological tripletpairing superconductivity^{6,7,8}, where electron fractionalizes into Majorana excitations^{9,10} and is the foundation for topological quantum computing^{11,12}. Therefore, pursuing such exotic superconductivity in realistic compounds constitutes a stimulating research topic.
The singleband Hubbard model, as the prototypical model carrying the strong correlation effects, has been widely employed in the studies of manyelectron systems^{13,14,15,16,17,18} as variants of this model are relevant to the twodimensional (2D) cuprate superconductors. Besides, quasi1D cuprate chains also constitute important class of strongly correlated materials that host intriguing correlated electron states and effects, e.g., the TomonagaLuttinger liquid (TLL) with spincharge separation^{19,20,21}. On the other hand, most theoretical studies of the groundstate and dynamical properties^{22,23,24,25} also lie in 1D as rigorous manybody simulations are more accessible using analytics^{26}, exact diagonalization^{27,28,29,30}, density matrix renormalization group (DMRG)^{31,32,33,34} and quantum Monte Carlo^{35,36,37,38,39,40,41}. Since both the onsite interaction U and nearneighbor (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 materials^{36,41,42,43,44,45}.
Most recently, a paradigm shift occurs as an in situ ARPES experiment on the 1D cuprate chain Ba_{2−x}Sr_{x}CuO_{3+δ} (BSCO) suggests an anomalously strong attraction V < 0 between NN electrons^{46}. In contrast to the intrinsic electronelectron Coulomb repulsion, this attractive interaction is likely to be mediated by electronphonon coupling^{47}. 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 positiveU and negativeV system in the transition metal oxides, where strong correlations and electronphonon coupling widely coexist^{48,49,50,51,52,53}. Such an effective attraction largely missed previously may serve as a key ingredient in both understanding the highT_{c} superconductivity and enabling exotic quantum phases in correlated materials^{27,28,29,30,54,55,56,57,58,59,60,61,62,63}. 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 attractiveV extended Hubbard model (EHM, see Fig. 1a), we employ largescale DMRG simulations and systematically explore its phase diagram. We especially focus on the possible realization of spintriplet superconductivity while identifying all phases. At both half and quarter fillings, we have numerically determined the groundstate phase diagrams of the EHM, from which we identify a robust gapless TLL phase with a prominent spintriplet superconducting (TS) pairing with algebraic singularity. In two dimensions, the triplet superconducting state is topologically nontrivial where the fractional excitation can emerge on the boundary^{10,64,65,66,67}. However, quantum fluctuations are usually too strong in 1D such that interacting electrons in a Hubbardtype chain usually behave as a TLL, contradicting the meanfield and smallcluster predictions. Therefore in this paper, we refer to this emergent TLL phase with divergent superconducting susceptibility as a gapless TS phase.
Our main findings are summarized in Fig. 1. At half filling (see Fig. 1b), the TS phase survives only up to a finite U_{c}/t ≃ 2.3 and is absent when U > U_{c}. At quarter filling (see Fig. 1c), this TS phase extends to larger Us comparable to those in cuprates^{46,68,69}. Between this TS phase and the regular phase separation (PS) phases with singly (PS_{1}) and doubly (PS_{2}) occupied clusters, we further identify an exotic PS_{x} phase where the clustered electrons form the TLL and even TS states. With the model parameters determined from fitting dynamical data of BSCO^{46}, our study reveals a close proximity of this doped cuprate chain to the pwave 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 onsite U > 0 and NN attraction V < 0^{46}, whose Hamiltonian reads
where \({c}_{i\sigma }^{{{{\dagger}}} }\) (c_{iσ}) is the electron creation (annihilation) operator, σ = ↑, ↓ labels the electron spin, and n_{i} = n_{i↑} + n_{i↓} 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 nonAbelian symmetry implemented^{70,71} (see “Methods” and Supplementary Note 1).
To characterize various quantum phases, we compute the spin, charge, and pairing correlation functions. The spinspin correlation is defined as F(r) = 〈S_{i} ⋅ S_{j}〉, with S_{i(j)} the spin operator at site i(j) and r ≡ j − i. The charge density correlation is defined as D(r) = 〈n_{i}n_{j}〉 − 〈n_{i}〉〈n_{j}〉, where n_{i(j)} is the particle number operator at site i(j). To characterize the superconducting (SC) pairing correlation, we consider both the spinsinglet (swave) 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 (pwave) 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 spintriplet 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 Φ_{S}.
Analytical results from the TLL theory
The TLL theory puts rigorous constraints^{72,73,74,75} on numerical results, which we always compare with and make use of in the analysis of our numerical data. In TLL, twopoint 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 K_{σ} and K_{ρ}, 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 momentumdependent spin structure factor S_{m}(k) and charge structure factor S_{c}(k), and then extract K_{σ} and K_{ρ}.
For the current EHM in Eq. (1) with SU(2) spin symmetry, K_{σ} = 1 for the spin density wave (SDW), TLL, and TS phases with gapless spin excitations, while K_{σ} = 0 in the spin gapped phase PS_{2}. Therefore, K_{ρ} uniquely determines the powerlaw exponents α of various correlations: for charge and spin correlations there exist a uniform mode with exponent α_{0} = 2 and a 2k_{F} mode with \({\alpha }_{2{k}_{{{{{{{{\rm{F}}}}}}}}}}=1+{K}_{\rho }\); for the pairing correlations Φ_{S} and Φ_{T}, they both have uniform modes with the same exponent α_{SC} = 1 + 1/K_{ρ}, which dominates over the spin and charge correlations when K_{ρ} > 1. Consequently, the lowT behaviors of the staggered magnetic, charge, and pairing susceptibilities are also controlled by K_{ρ}, 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 K_{ρ} > 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 K_{ρ} via a secondorder polynomial fitting of S_{c}(k) in the small k regime^{41,45,73,76} (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 PS_{2} 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.
The Luttinger parameter K_{ρ} clearly separates the U = 1.6 systems into three regimes. As the interaction strength increases to ∣V∣ > ∣V_{c}∣ ≃ 1 (but smaller than the phase separation transition strength ∣V_{s}∣, which will be discussed later), in Fig. 2a there exists an intermediate regime with K_{ρ} > 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 ∣V_{c}∣ < ∣V∣ < ∣V_{s}∣, 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 K_{ρ} 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∣ > ∣V_{s}∣. The critical strength V_{s} dependent on U is shown in Fig. 1 (see the detailed estimation of V_{s} in Supplementary Note 1). Specifically for the two selected cuts, we found V_{s} ≃ −1.55 for U = 1.6 (see Fig. 2a–c) and V_{s} ≃ − 2.42 for U = 4 (see Fig. 2d–f). In such a PS state, the clustered part consists of doublyoccupied sites and no singularity can be observed in various correlations. Therefore, we denote it as PS_{2} to distinguish from other PS phases discussed later.
Among these three phases in the U = 1.6 case (and for other interactions U < U_{c} ≃ 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 K_{ρ} 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 α_{SC} = 1 + 1/K_{ρ}— smaller than the algebraic exponent (1 + K_{ρ}) of both the charge and spin correlations when K_{ρ} > 1— and thus constitutes the dominant correlation in the charge2e channel, with an algebraically diverging pairing susceptibility χ_{SC}(T) for low temperature T.
In the weak attraction regime ∣V∣ < ∣V_{c}∣, K_{ρ} remains finite (< 1) in Fig. 2a due to the strong finitesize effects and a small charge gap for U = 1.6, which would converge to zero in the thermodynamic limit. K_{σ} = 1 due to the spin SU(2) symmetry. In Fig. 3b, an quasilong range spin order with an algebraic exponent of α_{SDW} = 1 appears, which has logarithmically diverging spin structure factor of S_{m}(k = π) (see Fig. 2c, f and the insets). This is well consistent with the SDW scenario with a finite charge gap and quasilong range spin order (see Supplementary Note 5). On the other hand, for the intermediate phase in Fig. 2c S_{m}(π) ceases to increase vs. L, as the 2k_{F} mode spin correlation decays faster than ~r^{−2} shown in Fig. 3b, which reveals a nondiverging 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 (Φ_{S}) and tripletpairing (Φ_{T}) exhibit powerlaw decay behaviors, and the latter with pwave pairing symmetry clearly dominates over the former with the swave pairing symmetry. This is clearly demonstrated in Fig. 4a, where the strengths of the two correlations Φ_{T}(r) and Φ_{S}(r) are compared at a fixed distance r = 20. Though two pairing correlations are comparable in the SDW regime, Φ_{T}(r) clearly surpasses Φ_{S}(r) once entering the intermediateV phase: the latter turns to decreasing, while Φ_{T}(r) keeps increasing and becomes over one order of magnitude greater than Φ_{S}(r).
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 Φ_{T} firstly decays exponentially in the SDW phase (the blue dots), then exhibits powerlaw behaviors for ∣V_{c}∣ < ∣V∣ < ∣V_{s}∣ (the red dots), and decays again exponentially for ∣V∣ > ∣V_{s}∣ (the gray dots). We notice there is virtually no uniform but only 2k_{F} mode in Φ_{S}, as reflected in the smooth curves Φ_{S}(r) × (−1)^{r−1} in Fig. 3d. For the gapless TS phase where we are most interested in, the dominance of Φ_{T} is reflected by the comparison of Fig. 3c and d: Φ_{T}(r) decays slower than r^{−2}, while Φ_{S}(r) decays faster than r^{−2} (Fig. 3d). More quantitatively, the ratio between these two pairing correlations ∣Φ_{T}(r)/Φ_{S}(r)∣ scales in power law \({r}^{{K}_{\rho }1}\), since the leading scaling in Φ_{T} and Φ_{S} is \(1/{r}^{1+1/{K}_{\rho }}\) and \(1/{r}^{{K}_{\rho }+1/{K}_{\rho }}\), respectively (see Supplementary Note 2). We present such a powerlaw scaling extracted from our DMRG simulations in Fig 4b. Therefore, in the intermediate regime the pairing correlation Φ_{T} dominates over Φ_{S} not only in magnitude but actually in longdistance 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 U_{c} ≃ 2.3, located in a much narrower regime. On the other hand, when compared to more recent studies^{62,77} where the phase diagrams are only schematic, here we pinpoint the numerically accurate phase boundaries with largescale DMRG calculations and reveal the predominant triplet quasilong range TS pairing relevant to the realistic cuprate chain BSCO, decades after such a TS instability was proposed^{29,54}.
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 K_{ρ} > 1 characterizes the intrinsic nature of the correlations and separates the U = 4 systems into four phases (see Fig. 5a). Particularly, for V < V_{c} ≃ −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 ∣Φ_{T}/Φ_{S}∣ and found its envelop increasing monotonically as ∣V∣ enhances and exceeding 1 for V < V_{c} (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.
Besides the TS regime, there are three different inhomogeneous PS phases, i.e., PS_{1}, PS_{x}, and PS_{2} in Fig. 1c, in the doped system. The realspace 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 PS_{x}. For small U, like U = 2, the system jumps from PS_{x} to PS_{2} at a second transition point. In contrast, this transition is preceded by a third PS phase for large U > U_{c} ≃ 2.3 (the same as that of half filling). Taking U = 4 as an example, PS_{x} firstly transits into an n(L/2) = 1 phase (denoted as PS_{1}), and then jumps into PS_{2} 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 PS_{1} and PS_{2} actually remain intact for other doping since they reflect the local energy relation between singly and doubly occupied states. The quantum manybody states in the clustered part of the three PS phases — PS_{1}, PS_{2}, and PS_{x} — only depends on the interaction parameters U and V.
The existence of the PS_{x} phase was missed in early studies on the same model^{29,54}, and the distinct feature of PS_{x} is the clustered electrons that constitute a TLL liquid with fractional filling. With x continuous tuned by V, the clustered part of PS_{x} can also become close to half filling in terms of density, i.e., x = 1. Nevertheless, it is distinct from that in the PS_{1} 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 K_{ρ} > 1 regime with significant TS pairing correlations in the clustered part of PS_{x}, showing the existence of gapless TS cluster in (at least part of) the PS_{x} phase (see more details in Supplementary Note 6).
Discussion
Our simulation is motivated by but not restricted to the recently extracted attractiveV extended Hubbard model for 1D cuprate BSCO from experiments^{46}. Previously, there were weakcoupling renormalization group (RG) and functional RG studies of the halffilled EHM that suggested the presence of TS pairing based on the fieldtheoretical analysis ^{54,55,61,63}. Besides, there were also ED and DMRG calculations of the phase diagram for halffilled or doped cases^{28,29,30,56,57,59,60,62}. The TS phase is characterized by dominant triplet superconducting pairing correlation, whose longdistance 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 onsite repulsive and nearneighbor attractive interactions. At both half and quarter fillings, we identify a prominent gapless TS phase with the pwave pairing induced by the attractive interactions. Different from the longrange order (hidden) assumption in the context of meanfield theory, the pwave superconducting order identified in this correlated 1D chain is quasilongranged: the triplet pairing correlation Φ_{T} decays as a powerlaw at long distance and presents as the dominant charge2e excitations in the gapless TLL, and specially, at half filling it dominates over the singlet pairing Φ_{S} 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 quasi1D 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 chains^{46,47}. Therefore, a quantitatively accurate phase diagram, especially with doping dependence, will be essential for upcoming 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 → ∞ limit^{78,79}. 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 K_{ρ} for a wide range of doping. As shown in Fig. 6, the uniform TS phase characterized as K_{ρ} > 1 can be realized only if ∣V∣ > 1.2 (and ∣V∣ ≲ 1.7 before PS_{x} 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 onsite U or enhancement of nearneighbor attraction V may drive it into the TS phase—both can be achieved by manipulating the electronphonon coupling either inside the crystal or via a substrate^{47}.
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 chain^{69}, 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 pwave 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 manybody studies of the superconductivity in the EHM of higher dimensions^{80,81,82,83}.
Lastly, our conclusion is not restricted to the cuprate BSCO. Considering the widely existing electron repulsion and electronphonon coupling^{48,49,50,51,52,53}, this model with a repulsive U and an attractive V may also be applicable, as a lowenergy approximation, for other transitionmetal 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 QSpace^{70,71}, 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).
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
All numerical codes in this paper are available upon request to the authors.
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Acknowledgements
We acknowledge Z. Chen, T.P. Devereaux, B. Moritz, Z.X. Shen, Yang Qi, T. Shi, and H.Q. Lin for stimulating discussions. W.L. acknowledges the support from the National Natural Science Foundation of China (Grant Nos. 12222412, 11974036, 11834014, and 12047503), and CAS Project for Young Scientists in Basic Research (Grant No. YSBR057). H.C.J. was supported by the Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract DEAC0276SF00515. Y.W. acknowledges support from U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Early Career Award No. DESC0022874. D.W.Q and W.L. thank the Highperformance Computing Center at ITPCAS for their technical support and generous allocation of CPU time.
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W.L. and Y.W. initiated this work. D.W.Q and B.B.C performed the DMRG calculations. All authors contributed to the analysis of the results. W.L., Y.W., and H.C.J. supervised the project.
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Qu, DW., Chen, BB., Jiang, HC. et al. Spintriplet pairing induced by nearneighbor attraction in the extended Hubbard model for cuprate chain. Commun Phys 5, 257 (2022). https://doi.org/10.1038/s4200502201030x
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DOI: https://doi.org/10.1038/s4200502201030x
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