Introduction

Quantum spin liquids (QSLs) are exotic phases of matter that exhibit a variety of novel features associated with their topological character1,2,3,4,5. The simplest QSLs known theoretically are characterized by topological order and support fractional excitations such as spinons, which carry the spin-1/2 but not the charge of the electron. The close relationship between the QSL and the superconductor suggests that theoretically doping it might naturally lead to superconductivity5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20. Indeed, this is the original dream of the resonating valence bond theory as a mechanism for high temperature superconductivity6. However, despite intense efforts during past three decades, definitive evidence showing that doping QSLs gives rise to superconductivity is lacking5,14,15,21,22,23,24. This is partially due to the fact that the realization of QSLs is a great challenge to physicists, where candidate materials and systems are rare2,3,4,5.

The spin-1/2 antiferromagnet on the triangular lattice with nearest-neighbor (NN) and next-nearest-neighbor (NNN) Heisenberg interactions is highly frustrated. A number of numerical simulations have provided strong evidence that its ground state is a QSL in the region of 0.07 ≤ J2/J1 ≤ 0.15, which is sandwiched by a 120° magnetic order phase and a stripe magnetic order phase25,26,27,28,29,30,31,32. Therefore, it can serve as an ideal platform to investigate the consequence of doping the QSL. Although it is still under debate whether this QSL is gapped or gapless in the 2D limit, it is consistent with a gapped (possibly “Z2”33) spin liquid on finite width cylinders. This is evidenced by the observed nonzero spin-gap and exponentially falling spin–spin correlations where the correlation length is significantly shorter than the width (Ly in Fig. 1) of the cylinders, and short-range dimer–dimer and chiral–chiral correlations25,26,27,28,29. Therefore, these leave little doubt that doping this state corresponds to doping a gapped QSL.

Fig. 1: The t-J model on the triangular lattice.
figure 1

Periodic and open boundary conditions are imposed respectively along the directions specified by the lattice basis vectors ey and ex. Lx and Ly are the number of sites in the ex and ey directions. t1 (t2) and J1 (J2) are hopping integral and spin exchange interactions between NN (NNN) sites. The electrons live at the vertices (filled circles), and a, b, and c denote the three different bonds.

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As the QSL has been realized on the spin-1/2 antiferromagnetic J1–J2 Heisenberg model on the triangular lattice, a natural question is whether doping this QSL will give rise to superconductivity? Quasi-one-dimensional (1D) systems such as cylinders have become an important starting point to resolve this problem which has essential degrees of freedom that allow for two-dimensional characteristics to emerge. Moreover, it can be accurately studied using the density-matrix renormalization group (DMRG)34,35,36. According to the Mermin–Wagner theorem, a superconducting (SC) state that can be realized in quasi-1D systems such as cylinders has quasi-long-range SC correlation. The Luther-Emery (LE) liquid is an example of this37, which has one gapless charge mode with quasi-long-range SC and charge-density-wave (CDW) correlations, but the spin–spin correlations fall exponentially38,39,40,41,42,43,44.

In this paper, we directly address the question of doping the QSL on the triangular lattice by studying the t-J model defined in Eq. (6). On the cylinders we study, the undoped system is a fully gapped QSL. Upon lightly doping we find that the system still shows exponentially falling spin–spin correlations. Most importantly, we find that even at the smallest doping level δ and on our largest 6-leg cylinders, the SC correlations are strong and decay with a slow power law, \({{\Phi }}(r) \,\sim\, | r{| }^{-{K}_{sc}}\), with Ksc ≈ 1. This slow decay implies a SC susceptibility that diverges as \({\chi }_{sc} \,\sim\, {T}^{-(2\,-\,{K}_{sc})}\) as the temperature T → 0. Moreover, the SC correlations dominate over the CDW correlations in the sense that in all cases Kc > Ksc. This is suggestive that doping the QSL state on the triangular lattice could naturally give rise to superconductivity.

Results

Principal results

We find that the system exhibits power-law CDW correlations corresponding to a local pattern of “partial-filled” charge stripes. For width Ly = 4 cylinders, the wavelength of the CDW, i.e., the spacings between two adjacent charge stripes, is λ = 1/2δ corresponding to an ordering wavevector Q = 4πδ with half a doped hole per unit cell (Fig. 2a). Equivalently, if we view the cylinder as a 1D system, this corresponds to two holes per 1D unit cell. For width Ly = 6 cylinders, the CDW has a wavelength of λ = 1/3δ, and an ordering wavevector Q = 6πδ with only one third of a doped hole per unit cell (Fig. 2b). Similar with the 4-leg cylinders, this again corresponds to two holes per 1D unit cell.

Fig. 2: Charge-density profiles and Luttinger exponent Kc.
figure 2

Charge-density profiles n(x) for (a) Ly = 4 and (b) Ly = 6 cylinders of length Lx at various doping levels δ. The extracted Kc using Eq. (1) for (c) Ly = 4 and (d) Ly = 6 cylinders as a function of 1/Lx and δ, as well as their extrapolated value in the long cylinder limit 1/Lx = 0, i.e., Lx → ∞. A few data points in light gray are removed to minimize the boundary effect. Error bars denote the numerical uncertainty and the dash-dotted lines denote the linear extrapolation.

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Because Lx ≫ Ly, our system can be thought of as quasi-1D, and we find that the ground state of the system is consistent with that of a LE liquid37, which is characterized by power-law decaying CDW (Fig. 2) and SC correlations (Figs. 3, 4), but exponentially decaying spin–spin correlations (Fig. 5), with one gapless charge mode (Fig. 6). At long distance, the spatial decay of the CDW correlation is dominated by a power-law with the Luttinger exponent Kc. The exponent Kc can be obtained by fitting the charge-density oscillations (Friedel oscillations) induced by the boundaries of the cylinder45

$$n(x)\,=\,{n}_{0}\,+\,\delta n* \cos (2{k}_{F}x\,+\,\phi ){x}^{-{K}_{c}/2}.$$
(1)

Here \(n(x)\,=\,\langle \hat{n}(x)\rangle\), where \(\hat{n}(x)\,=\,\mathop{\sum }\nolimits_{y \,=\, 1}^{{L}_{y}}\hat{n}(x,y)/{L}_{y}\) is the local rung density operator and x is the rung index of the cylinder. δn is a nonuniversal amplitude, ϕ is a phase shift, n0 is the background density, and kF is the Fermi wavevector. Note that a few data points (Fig. 2a, b, gray color) are excluded to minimize the boundary effect and improve the fitting quality. The extracted exponent Kc is similar on both width Ly = 4 and Ly = 6 cylinders, where we find Kc > 1 at all doping levels. Similarly, Kc can also be obtained from the charge density-density correlation which gives consistent results (see Supplementary Fig. 4).

Fig. 3: Superconducting correlations on Ly = 4 cylinders.
figure 3

SC correlations Φaa(r) on Ly = 4 cylinders of length Lx for (a) δ = 1/12, (b) δ = 1/16, and (c) δ = 1/24 on double-logarithmic scales, where r is the distance between two Cooper pairs in the ex direction. The solid lines denote power-law fitting \({{{\Phi }}}_{aa}(r) \,\sim\, {r}^{-{K}_{sc}}\). d Luttinger exponent Ksc as a function of δ and 1/Lx, as well as their extrapolated value in the long cylinder limit 1/Lx = 0, i.e., Lx → ∞. Inset: The extrapolated exponents Ksc (1/Lx = 0), Kc(1/Lx = 0) in the long cylinder limit 1/Lx = 0 and their product KscKc as a function of δ. Error bars denote the numerical uncertainty and the dash-dotted lines denote the linear extrapolation.

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Fig. 4: Superconducting correlations on Ly = 6 cylinders.
figure 4

SC correlations Φaa(r) on Ly = 6 cylinders of length Lx for (a) δ = 1/12, (b) δ = 1/18, and (c) δ = 1/24 on double-logarithmic scales, where r is the distance between two Cooper pairs in the ex direction. The solid lines denote power-law fitting \({{{\Phi }}}_{aa}(r) \,\sim\, {r}^{-{K}_{sc}}\). d Luttinger exponent Ksc as a function of δ and 1/Lx, as well as their extrapolated value in the long cylinder limit 1/Lx = 0, i.e., Lx → ∞. Inset: The extrapolated exponents Ksc (1/Lx = 0), Kc(1/Lx = 0) in the long cylinder limit 1/Lx = 0 and their product KscKc as a function of δ. Error bars denote the numerical uncertainty and the dash-dotted lines denote the linear extrapolation.

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Fig. 5: Spin–spin correlations.
figure 5

a Examples of spin–spin correlation functions F(r) at doping level δ = 1/12 for Ly = 4 and Ly = 6 cylinders on the semi-logarithmic scale. Solid lines denote exponential fitting \(F(r) \,\sim\, {e}^{-r/{\xi }_{s}}\), where r is the distance between two sites in the ex direction. b The spin–spin correlation length ξs as a function of δ for both Ly = 4 and Ly = 6 cylinders. Error bars denote the numerical uncertainty.

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Fig. 6: Entanglement entropy S and central charge c.
figure 6

S is shown at doping level δ = 1/12 for (a) Ly = 4 and (b) Ly = 6 cylinders. Extracted central charge c using Eq. (5) for (c) Ly = 4 and (d) Ly = 6 cylinders a a function of Lx and δ. Note that a few data points in blue from each end are removed to minimize boundary effects. Error bars denote the numerical uncertainty.

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At long distance, the SC pair-field correlation Φ(r), defined in Eq. (3), is also characterized by a power-law with the appropriate Luttinger exponent Ksc defined by

$${{\Phi }}(r)\,\propto\, {r}^{-{K}_{sc}},$$
(2)

where r is the distance between two Cooper pairs along the ex direction. As shown in Fig. 3d, the extracted exponent in the long cylinder limit Lx → ∞ is Ksc < 1 on Ly = 4 cylinders. This also holds true for Ly = 6 cylinders, again we find that Ksc < 1 when Lx → ∞. For both cases, this implies a divergent SC susceptibility \({\chi }_{sc} \,\sim\, {T}^{-(2\,-\,{K}_{sc})}\) when the temperature T → 0. As expected theoretically from the LE liquid37, we find that the relation KcKsc = 1 holds within the numerical uncertainty and finite-size effect (Figs. 3d, 4d, insets) for Ly = 4 cylinders for doping levels δ ≤ 1/12 and Ly = 6 cylinders for doping levels δ ≤ 1/18. However, a notable deviation can be seen for the doping level δ = 1/12 on Ly = 6 cylinders (Fig. 4d inset), we suspect that it might be attributed to the finite-size effect even after finite-size (Lx) extrapolation (Figs. 2d, 4d). This is also consistent with the central charge c shown in Fig. 6d, which also suffers from visible finite-size effect, although an apparent trend that c may approach to the expected value c = 1 in the long cylinder limit Lx → ∞. In any case, the fact that Kc > Ksc (Figs. 3d, 4d) demonstrates the dominance of the SC correlations Φ(r) on both Ly = 4 and Ly = 6 cylinders.

Charge-density wave order

To describe the charge-density properties of the ground state of the system, we have calculated the charge-density profile n(x). Figure 2a shows the charge-density distribution n(x) for width Ly = 4 cylinders, which is consistent with the “half-filled” charge stripe of wavelength λ = 1/2δ, i.e., spacings between two adjacent stripes, for all doping concentration δ with half a doped hole per unit cell. The charge-density profile n(x) for width Ly = 6 cylinders is shown in Fig. 2b, which is however consistent with “third-filled” charge stripes for all δ with only one third of a doped hole per unit cell. As a result, it has a shorter wavelength λ = 1/3δ on Ly = 6 cylinders than the "half-filled" stripes on Ly = 4 cylinders at the same doping concentration δ.

At long distance, we find that the CDW correlations decay with a power-law with the Luttinger exponent Kc. The exponent Kc, which was obtained by fitting the data points using Eq. (1), is shown in Fig. 2c for width Ly = 4 cylinders and Fig. 2d for width Ly = 6 cylinders as a function of Lx and δ. For all cases, we find that Kc increases with Lx, which is consistent with weaker CDW correlations in the longer cylinders. A notable difference between the Ly = 4 and Ly = 6 cylinders is the doping dependence of Kc as shown in Figs. 3d, 4d. On the Ly = 4 cylinders, the value of Kc decreases with the increase of δ on width Ly = 4 cylinder. On the contrary, Kc increases with the increase of δ on width Ly = 6 cylinders. However, in any case, our results show that Kc > 1, or more precisely Kc > 1.2 in the long cylinder limit Lx → ∞ (Fig. 2c, d), for all doping levels on both Ly = 4 and Ly = 6 cylinders. Kc can also be obtained directly from the charge-density-density correlation, which produces similar results (see Supplementary Fig. 4).

Superconducting correlation

To test the possibility of superconductivity, we have also calculated the equal-time SC pair-field correlations. As the ground state of the system with even number of doped holes is always found to have spin 0, we focus on spin-singlet pairing. A diagnostic of the SC order is the pair-field correlator, defined as

$${{{\Phi }}}_{\alpha \beta }(r)\,=\,\frac{1}{{L}_{y}}\mathop{\sum }\limits_{y\,=\,1}^{{L}_{y}}\,\langle {{{\Delta }}}_{\alpha }^{{\dagger} }({x}_{0},y){{{\Delta }}}_{\beta }({x}_{0}\,+\,r,y)\rangle .$$
(3)

Here \({{{\Delta }}}_{\alpha }^{{\dagger} }(x,y)\,=\,\frac{1}{\sqrt{2}}[{c}_{(x,y),\uparrow }^{{\dagger} }{c}_{(x,y)\,+\,\alpha ,\downarrow }^{{\dagger} }\,-\,{c}_{(x,y),\downarrow }^{{\dagger} },{c}_{(x,y)\,+\,\alpha ,\uparrow }^{{\dagger} }]\) is the spin-singlet pair-field creation operator, where the bond orientations are designated α = a, b, and c (Fig. 1), (x0, y) is the reference bond with x0 ~ Lx/4, and r is the distance between two bonds in the ex direction.

Due to the presence of CDW modulations (Fig. 2), SC correlations Φαβ(r) exhibit similar spatial oscillations with n(x). For a given cylinder of length Lx, we determine the decay of SC correlations with reference bond located at the peak position around x0 ~ Lx/4 of the charge-density distribution n(x) to minimize the boundary effect42,43. This gives accurate values of Φαβ(r) for reliable finite-size scaling. Examples of Φaa(r) are given in Figs. 3, 4 for width Ly = 4 and Ly = 6 cylinders, respectively. Further details are provided in the Supplementary Fig. 2.

As the ground state of the system is a spin-singlet state, theoretically, the pair symmetry of the SC correlations will be consistent with either s-wave, d ± id or d-wave46,47. In the following we show that the pair symmetry is only consistent with the d-wave. First of all, our results show that the ground state wavefunction of the system is purely real and the imaginary part of all the SC correlations Φαβ(r) is zero. As a result, the pair symmetry is inconsistent with d ± id-wave. Second, the SC correlations change sign between different bonds, i.e., Φab ~ Φac ~ −0.4Φaa (see Supplementary Fig. 3), so the pair symmetry is also inconsistent with s-wave. Therefore, the only option left is the d-wave, which is expected to survive in two dimensions47. This is indeed consistent with our results, where we find that the SC correlations are stronger on a bonds but weaker on other bonds, i.e., Φbb ~ Φcc ~ 0.2Φaa on both the Ly = 4 and Ly = 6 cylinders. Taken together theory and numerical results, we therefore conclude that the pairing symmetry of the SC correlations is consistent with the d-wave.

Similar to the CDW correlations, the SC correlations Φ(r) of the system on both width Ly = 4 (Fig. 3) and Ly = 6 cylinders (Fig. 4) also decay with a power law, whose exponent Ksc was obtained by fitting the results using Eq. (2)42,43,44. The fact that the observed Ksc < 1 holds for all the cases when Lx ≫ Ly undoubtedly demonstrates the dominance of the SC correlations over the CDW correlations since Kc > 1 > Ksc. These findings are in stark contrast to the case of doping QSL on the Kagome lattice22,24, where the SC correlations decay exponentially but the CDW correlations have 2D long-range order. This indicates that the lattice geometry may play a critical role in giving rise to superconductivity in doping a QSL.

Spin–spin correlation

To describe the magnetic properties of the ground state, we calculate the spin–spin correlation functions defined as

$$F(r)\,=\,\frac{1}{{L}_{y}}\mathop{\sum }\limits_{y\,=\,1}^{{L}_{y}}| \langle {\overrightarrow{S}}_{{x}_{0},y}\,\cdot\, {\overrightarrow{S}}_{{x}_{0}\,+\,r,y}\rangle | ,$$
(4)

where \({\overrightarrow{S}}_{x,y}\) is the spin operator on site i = (x, y). (x0, y) is the reference site with x0 ~ Lx/4 and r is the distance between two sites in the ex direction. It is worth noting that the oscillations have different wavelength between small r region (short distance) and large r region (long distance), and there are no antiphase domain walls in the spin–spin correlation \(\langle {\overrightarrow{S}}_{{x}_{0},y}\,\cdot\, {\overrightarrow{S}}_{{x}_{0}\,+\,r,y}\rangle\). Following similar procedure as for n(x) and Φ(r), we first extrapolate F(r) to the limit ϵ → 0 for a given cylinder of length Lx and then perform finite-size scaling as a function of Lx. As shown in Fig. 5a, F(r) decays exponentially as \(F(r) \,\sim\, {e}^{-r/{\xi }_{s}}\) at long-distances, with a correlation length ξs = 1 ∼ 8 lattice spacings for the Ly = 4 cylinders and ξs = 2 ~ 7 lattice spacings for the Ly = 6 cylinders (Fig. 5b) for doping concentration 0 ≤ δ ≤ 1/12. Therefore, the spin–spin correlations are also short-ranged which is similar with the QSL at half-filling, where the spin–spin correlations also decay exponentially.

Central charge

A key feature of the LE liquid is that it has a single gapless charge mode with central charge c = 1, which can be obtained by calculating the von Neumann entropy \(S\,=\,-{{{\rm{Tr}}}}\rho ln\rho\), where ρ is the reduced density matrix of a subsystem with length x. For critical systems in 1 + 1 dimensions described by a conformal field theory, it has been established48,49 that for an open system of length Lx,

$$\begin{array}{lll}S(x)&=&\frac{c}{6}{{\mathrm{ln}}}\,\left[\frac{4({L}_{x}\,+\,1)}{\pi }\sin \frac{\pi (2x\,+\,1)}{2({L}_{x}\,+\,1)}| \sin {k}_{F}| \right]\\ &&+\,\tilde{A}\frac{\sin [{k}_{F}(2x\,+\,1)]}{\frac{4({L}_{x}\,+\,1)}{\pi }\sin \frac{\pi (2x\,+\,1)}{2({L}_{x}\,+\,1)}| \sin {k}_{F}| }\,+\,\tilde{S},\end{array}$$
(5)

where \(\tilde{A}\) and \(\tilde{S}\) are model dependent fitting parameters, and kF is the Fermi momentum. Examples of S(x) are shown in Fig. 6a, b for the Ly = 4 and Ly = 6 cylinders, respectively. For a given cylinder of length Lx, a few data points in blue close to the ends are excluded in the fitting to minimize the boundary effect. The extracted central charge c is shown in Fig. 6c, d as a function of Lx and δ. Although the value of c deviates notably from c = 1 on short cylinders, apparently it approaches c = 1 with the increase of Lx by taking into account the numerical uncertainty and finite-size effect. The result is consistent with one gapless charge mode with c = 1, which provides additional evidence for the presence of the LE liquid in doping the QSL on the triangular lattice.

Discussion

Taken together, our results show that the ground state of doping QSL on the triangular lattice is consistent with a LE liquid with a finite gap in the spin sector. Although both CDW and SC correlations decay as a power-law, the fact that Kc > 1 > Ksc on both the 4-leg and 6-leg cylinders when Lx ≫ Ly at all doping levels undoubtedly demonstrates the dominance of the SC correlations with divergent SC susceptibility. To the best of our knowledge, this is the direct numerical realization of dominant superconductivity in doping a time-reversal symmetric QSL, which provides compelling evidence that doping a QSL could naturally give rise to superconductivity6,7,8,9. In this paper, we have focused on the lightly doped case, it will also be interesting to study the higher doping case as well as the consequence of doping different phases on the triangular lattice such as the magnetic ordered phase50,51. Answering these questions may lead to better understanding of the mechanism of high temperature superconductivity.

Methods

Model Hamiltonian

We employ DMRG34 to study the ground state properties of the hole-doped t-J model on the triangular lattice, defined by the Hamiltonian

$$H\,=\,-\mathop{\sum}\limits_{ij\sigma }{t}_{ij}\left({\hat{c}}_{i\sigma }^{+}{\hat{c}}_{j\sigma }\,+\,h.c.\right)\,+\,\mathop{\sum}\limits_{ij}{J}_{ij}\left({\overrightarrow{S}}_{i}\,\cdot\, {\overrightarrow{S}}_{j}\,-\,\frac{{\hat{n}}_{i}{\hat{n}}_{j}}{4}\right).$$
(6)

Here \({\hat{c}}_{i\sigma }^{+}\) (\({\hat{c}}_{i\sigma }\)) is the electron creation (annihilation) operator on site i = (xi, yi) with spin σ. \({\overrightarrow{S}}_{i}\) is the spin operator and \({\hat{n}}_{i}\,=\,{\sum }_{\sigma }{\hat{c}}_{i\sigma }^{+}{\hat{c}}_{i\sigma }\) is the electron number operator. The electron hopping amplitude tij is equal to t1 (t2) if i and j are NN (NNN) sites as shown in Fig. 1. J1 and J2 are the spin superexchange interactions between NN and NNN sites, respectively. The Hilbert space is constrained by the no-double occupancy condition, ni ≤ 1. At half-filling, i.e., ni = 1, Eq. (6) reduces to the spin-1/2 antiferromagnetic J1–J2 Heisenberg model.

Numerical details

The lattice geometry used in our simulations is depicted in Fig. 1, where ex = (1, 0) and \({{{{\bf{e}}}}}_{y}\,=\,(1/2,\sqrt{3}/2)\) denote the two basis vectors. We take the lattice geometry to be cylindrical with periodic (open) boundary condition in the ey (ex) direction. Here, we focus on cylinders with width Ly and length Lx, where Lx and Ly are number of sites along the ex and ey directions, respectively. The total number of sites is N = Lx × Ly with Ne = N electrons at half-filling. The doping level of the system is defined as δ = Nh/N, where Nh = N − Ne is the number of doped holes.

For the present study, we focus on the lightly doped case with δ ≤ 1/12 on width Ly = 4 cylinders of length up to Lx = 144 and width Ly = 6 cylinders of length up to Lx = 72. We set J1 = 1 as an energy unit and J2 = 0.11 such that the system is deep in the QSL phase at half-filling25,26,27,28,29,30. We consider t1 = 3 and \({t}_{2}\,=\,{t}_{1}\sqrt{{J}_{2}/{J}_{1}}\,=\,0.995\) to make a connection to the corresponding Hubbard model52. The results also hold for other t1 and t2. We perform up to 100 sweeps and keep up to m = 12,000 states for width Ly = 4 cylinders with a typical truncation error ϵ < 3 × 10−7, and up to m = 30000 states for width Ly = 6 cylinders with a typical truncation error ϵ < 4 × 10−6. This leads to excellent convergence for our results when extrapolated to m = ∞ limit. Further details of the numerical simulation are provided in the Supplementary Information sections I, II, and III.