Introduction

The pair-density wave (PDW) is a superconducting (SC) state in which the order parameter varies periodically in space in such a way that its spatial average vanishes1,2. The first example of PDW is the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state3,4 when a Zeeman magnetic field, H, is applied to a s-wave superconductor so that the Fermi surface is spin-split. The SC order has a wavevector Q ~ μBH/EF, which is typically very small. The LO version of this state is accompanied by an induced magnetization density wave and a charge-density wave (CDW) with ordering wavevectors K = 2Q. Intense interest in a somewhat different sort of PDW state has emerged due to recent discoveries in underdoped cuprate superconductors, where a direct observation of PDW has been made experimentally via local Cooper pair tunneling and scanning tunneling microscopy in Bi2Sr2CaCu2O8+x5,6,7, as well as the dynamical inter-layer decoupling observed in 1/8 hole-doped La2BaCuO48,9. While similar in having oscillatory SC order and associated K = 2Q CDW order, this PDW is conjectured to be stable in zero magnetic field (zero net magnetization), have an ordering vector that is independent of H (at least for small or vanishing H), and moreover can either have no associated magnetic order, or possibly have spin-density wave order (SDW) with the same ordering vector Q.

Although much is known about the properties of the PDW state1,2,10 there are very few microscopic models, which are shown to have PDW ground states. These include the one-dimensional (1D) Kondo-Heisenberg model with 1D electron gas coupled to a spin chain11, the extended two-leg Hubbard–Heisenberg model12, and strong coupling limit of the Holstein–Hubbard Model13,14. The evidence of PDW is also observed in the t-J model with four-spin ring exchange on a four-leg triangular lattice15 and an extended Hubbard model with a staggered spin-dependent magnetic flux per plaquette on a three-leg triangular lattice16. However, there is no evidence of PDW ordering found in more standard models even with second neighbor interactions17,18,19,20.

Theoretically, it has been proposed that superconductivity can also emerge in doping quantum spin liquids (QSLs), which are exotic phases of matter that exhibit a variety of features associated with their topological character21,22,23. The QSLs can be viewed as insulating phases with preexisting electron pairs such that upon light doping they might automatically yield high-temperature superconductivity24,25,26,27,28,29,30,31. Indeed, recent numerical studies using density-matrix renormalization group (DMRG) have provided strong evidences that lightly doping the QSL and chiral spin liquid on the triangular lattice will naturally give rise to nematic d-wave32 and d ± id-wave superconductivity33, respectively. Although dominant SC correlations have been observed in both systems, there is still no evidence for PDW ordering.

In this paper, we define a t-J-like extension of the Kitaev model on the honeycomb lattice (Fig. 1) so as to address the question of whether SC emerges upon light doping. The Kitaev model, i.e., JK term in Eq. (4), is exactly solvable and has a gapless spin liquid as its exact ground state34. Moreover, it has potential experimental realizations in magnets with strong spin-orbit coupling such as Na2IrO3 and α-RuCl335,36,37,38,39,40. This provides us a unique theoretical opportunity to test the physics of doping QSLs and may also give us some hints for understanding the mechanism of high-temperature superconductivity in the cuprates. Theoretically, doping Kitaev spin liquid (KSL) has been studied and distinct metallic states were proposed41,42,43,44,45. These include the p-wave superconductivity41,42, d-wave superconductivity43, topological superconductivity44, and Fermi liquid state45. However, controlled results of the sort that can be obtained using density-matrix renormalization group (DMRG) are still lacking concerning the phase(s) that arise upon doping the KSL.

Fig. 1: The schematic three-leg cylinder on the honeycomb lattice.
figure 1

The open (filled) circle denotes A (B) sub-lattice, and x, y, and z label the three different bonds. Periodic (open) boundary condition is imposed along the direction specified by the lattice basis vector e2 (e1). Lx (Ly) is the number of unit cells in the e1 (e2) direction.

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Results

Principal results

In the present paper, we study the lightly doped Kitaev model in Eq. (4) on the honeycomb lattice using DMRG46. Based on DMRG calculations on three-leg cylinders, we find that upon light-doping the KSL state, the system exhibits power-law CDW correlations at long distances corresponding to a local pattern of partially filled charge stripes. For three-leg cylinders, the wavelength of CDW, i.e., the spacings between two adjacent charge strips in the e1 direction, is λc = a0/3δ, where a0 is the length of unit cell. This corresponds to an ordering wavevector K = 3πδ/a0 with two thirds of a doped hole per CDW unit cell.

We find that the even-parity SC correlations are the most pronounced SC correlations, far dominant compared with the odd-parity SC or topological superconductivity41,42,44. Moreover, the dominant pairing channel oscillates in sign as a function of distance, which is consistent with the striped PDW2. Its wavelength λsc = 2a0/3δ is the same as that of the SDW correlations λs = 2a0/3δ and two times of that of the CDW λc = a0/3δ. Correspondingly, the SC ordering wavevector Q = 3πδ/2a0 is half of the ordering wavevector K = 3πδ/a0 of the CDW. However, our numerical results are inconsistent with the theoretical predictions39,40,41,42,43 as we find that both the single-particle Green functions and SC correlations are short-ranged, which decay exponentially at long distances. Although quasi-long-range SC correlations are not seen in the range of parameters we have studied, short-range correlations are fairly strong with the corresponding correlation length ξsc ≥ 3a0. This is comparable with the cylindrical width and is notably larger than that of doping the Kagome QSL47. Similar with doping the QSL on the triangular lattice32, we find that the pairing symmetry of SC correlations is also consistent with d-wave.

Superconducting correlations

To test the possibility of superconductivity, we have calculated the equal-time SC pair–pair correlations. A diagnostic of the SC order is the SC pair–pair correlation function, defined as

$${{{\Phi }}}_{\alpha \beta }(r)=\langle {{{\Delta }}}_{\alpha }^{\dagger }({x}_{0},y){{{\Delta }}}_{\beta }({x}_{0}+r,y)\rangle ,$$
(1)

where \({{{\Delta }}}_{\alpha }^{\dagger }(x,y)=\frac{1}{\sqrt{2}}[{\hat{c}}_{(x,y),\uparrow }^{\dagger }{\hat{c}}_{(x,y)+\alpha ,\downarrow }^{\dagger }-{\hat{c}}_{(x,y),\downarrow }^{\dagger }{\hat{c}}_{(x,y)+\alpha ,\uparrow }^{\dagger }]\) is the even-parity SC pair-field creation operator, where the bond orientations are labeled as α = x, y, z (Fig. 1). (x0, y) is the reference bond with \({x}_{0} \sim {\tilde{L}}_{x}/4\) to minimize the boundary effect and r is the distance between two bonds in the e1 direction. We have also calculated the odd-parity SC correlations to test the possibility of p-wave or topological superconductivity. However, we find that they are much weaker than the even-parity SC correlations (see Supplementary Discussion for details), suggesting that p-wave or topological superconductivity is unlikely. Therefore, we will focus on the even-parity SC correlation in this paper.

 Figure 2 shows the SC pair–pair function Φyy(r) for doping δ = 1/12 and 1/9. The SC correlation shows clear spatial oscillation for both doping levels, which can be well fitted by Φyy(r) ~ f(r) * ϕyy(r) for a large region of r as we will discuss below. Here, f(r) is the envelope function and ϕyy(r) is a spatial oscillatory function. At long distances, the envelope function f(r) is consistent with an exponential decay, i.e., \(f(r) \sim {{\rm{e}}}^{-r/{\xi }_{{\rm{sc}}}}\), as shown in Fig. 2a, b. The extracted correlation length is ξsc ≥ 3a0, which is comparable with the cylindrical width Ly = 3a0. Alternatively, the SC correlation at long distances can also be fitted by a power law (see Supplementary Discussion for details), i.e., \(f(r) \sim {r}^{-{K}_{{\rm{sc}}}}\), with an exponent Ksc > 4 for both doping levels δ = 1/12 and 1/9. We have also measured other types of SC correlations Φαβ(r), which are provided in the Supplementary Discussion. While they are slightly weaker than Φyy(r) due to the broken symmetry induced by the cylindrical geometry, they have very similar decaying behavior with Φyy(r). Although the SC correlations can be fitted by either functions, it is clear that the SC susceptibility will not diverge in the thermodynamic limit.

Fig. 2: Superconducting correlations and their spacial oscillation.
figure 2

Superconducting correlations ∣Φyy(r)∣ on three-leg cylinder at doping level a δ = 1/9 and b δ = 1/12, where dashed lines denote fittings to an exponential function \(f(r) \sim {{\rm{e}}}^{-r/{\xi }_{{\rm{sc}}}}\). Insets are the ratio Φxy(r)/Φyy(r) < 0, where the out of phase indicates d-wave type pairing. c The normalized function ϕyy(r) = Φyy(r)/f(r) at δ = 1/9 and δ = 1/12, which directly reflects the spatial oscillation of Φyy(r). d Fourier transformation ϕyy(q) of ϕyy(r) at δ = 1/12 and δ = 1/9, where peaks lie at Q = 3πδ/2a0 giving the total momentum of the pairing. Note that filled (open) symbols denote positive (negative) value.

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The spatial oscillation of the SC correlation Φyy(r) is characterized by the normalized correlation ϕyy(r) = Φyy(r)/f(r), as shown in Fig. 2c. It is clear that ϕyy(r) varies periodically in real space and can be well fitted by \({\phi }_{yy}(r) \sim \sin\)(Qr + θ) for both cases. This is consistent with the PDW state with vanishing spatial average of ϕyy(r). The ordering wavevector of the PDW is Q = 3πδ/2a0 as indicated by the peak position of the Fourier transformation ϕyy(q) of ϕyy(r) (Fig. 2d) and θ is a fitting phase factor. The corresponding wavelength is λsc = 2a0/3δ, which is λsc = 8a0 for δ = 1/12 and λsc = 6a0 for δ = 1/9. As we will see below that the relation λsc = λs = 2λc can be clearly seen in our results as expected from that of striped PDW. Here, λc and λs are the wavelengths of the CDW and SDW, respectively. According to Ginzburg-Landau theory, the development of K = 2Q charge oscillation corresponds to the cubic term \({\rho }_{K}{{{\Delta }}}_{Q}^{* }{{{\Delta }}}_{-Q}\) in the free energy, where ρK and ΔQ are the charge density and PDW order parameters with corresponding momenta K and Q, respectively. This is distinct from the CDW modulated superconductivity with term \({\rho }_{Q}{{{\Delta }}}_{0}^{* }{{{\Delta }}}_{-Q}\) with coexisting dominant uniform Δ0 and secondary stripe pairing ΔQ.

To identify the pairing symmetry, we have calculated the SC correlations using both real-valued and complex-valued DMRG simulations. We first rule out the d ± id-wave symmetry as we find that both the wavefunction and SC correlations are real while their imaginary parts are zero. We have further analyzed the relative phase of different SC correlations, e.g., Φxy(r)/Φyy(r) in the insets of Fig. 2a, b, where a clear out of phase can be observed as Φxy(r)/Φyy(r) < 0. Therefore, we conclude that the pairing symmetry of the PDW is consistent with d-wave.

Charge-density wave

In addition to SC correlations, we have also measured the charge-density profiles to describe the charge-density properties of the system. The rung charge density \(n(x)=\mathop{\sum }\nolimits_{y = 1}^{{L}_{y}}n(x,y)/{L}_{y}\) is shown in Fig. 3, where x is the rung index of the cylinder and n(x, y) is the local charge-density on site i = (x, y). It is clear that the charge density varies periodically in real space along the e1 direction with the wavelength λc = a0/3δ, i.e., λc = 4a0 and λc = 3a0 for δ = 1/12 and δ = 1/9, respectively. Therefore, there are two thirds of a doped hole in each CDW unit cell. Moreover, it is clear that the relation λsc = 2λc holds for both δ = 1/12 and δ = 1/9, which is consistent the striped PDW state. Different with SC correlations, we find power-law decay of the charge-density correlation at long distances. The Luttinger exponent Kc of the power-law decay can be extracted by fitting the charge-density oscillation (Friedel oscillation) induced by the open boundaries of the cylinder48,49

$$n(x)={n}_{0}+{\rm{\delta n}}* \cos (Kx+\theta ){x}^{-{K}_{{\rm{c}}}/2}.$$
(2)

Here, x is the distance in the e1 direction from the open boundary, n0 the average density and K is the ordering wavevector. δn and θ are the model-dependent constants. Examples of the fitting using Eq. (2) on the a sub-lattice are given in Fig. 3a for both doping δ = 1/12 and δ = 1/9, where four data points near the boundary are removed to minimize boundary effect for a more reliable fit. The extracted exponent Kc is given in Table 1. It is clear that Kc < 1, which demonstrates the dominance of the charge-density correlations.

Fig. 3: CDW and Luttinger exponent.
figure 3

a Charge-density profiles n(x) for three-leg cylinder of length Lx = 48 at doping levels δ = 1/9 and δ = 1/12 on A and B sub-lattices, respectively. The solid lines denote the fitting using Eq. (2). b The extracted exponent Kc at δ = 1/9 as a function of truncation error ϵ. The dashed line denotes Kc extracted from Acdw(Lx) in (d). c Convergence and length dependence of the CDW amplitude Acdw for three-leg cylinder at δ = 1/9 as a function of truncation error ϵ. The solid lines denote fitting using second-order polynomial function. d Finite-size scaling of Acdw(Lx) as a function of Lx in a double-logarithmic plot at δ = 1/9, where the dashed line denotes the fitting to a power law \({A}_{{\rm{cdw}}}({L}_{x}) \sim {L}_{x}^{-{K}_{{\rm{c}}}/2}\).

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Table 1 Summarized parameters.
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Alternatively, we can estimate Kc from the amplitude Acdw(Lx) of the charge-density modulation19. For a given cylinder of length Lx, the CDW amplitude Acdw(Lx) can be obtained by fitting the central-half region of the charge-density profile n(x)18,19,20. For quasi-long-range charge order, the amplitude should follow \({A}_{{\rm{cdw}}}({L}_{x})\propto {L}_{x}^{-{K}_{{\rm{c}}}/2}\) with similar Kc with that obtained from Friedel oscillation in Eq. (2). This is indeed the case as shown in Fig. 3c, d, where the exponent Kc is given in Fig. 3b and Table 1.

Spin–spin and single-particle correlations

To describe the magnetic properties of the system, we have further calculated the spin–spin correlation function \({F}_{\gamma }(r)=\langle {S}_{{i}_{0}}^{\gamma }{S}_{{i}_{0}+r}^{\gamma }\rangle\). Here, r is the distance between two sites in the e1 direction and γ = x, y, z. i0 = (x0, y) is the reference site with \({x}_{0} \sim {\tilde{L}}_{x}/4\). For the pure Kitaev model without doping, it is known that Fγ(r) is nonzero only for NN sites34,50. Upon doping, the Z2 flux on each hexagonal plaquette is no longer a conserved quantity, and the spin–spin correlation functions become nonzero even at long distance. For both doping levels δ = 1/12 and δ = 1/9 on three-leg cylinders, we find that both Fx(r) and Fz(r) decay exponentially as \(F(r) \sim {{\rm{e}}}^{-r/{\xi }_{{\rm{s}}}}\), where the spin–spin correlation length is given in Table. 1. On the contrary, Fy(r) appears to be long-range ordered as shown in Fig. 4b. This is unexpected but allowed theoretically since there is no continuous spin symmetry in the system. Moreover, we find that the spin–spin correlation Fy(r) shows clear spatial oscillation as shown in the inset of Fig. 4a with a wavelength λs that is the same as that of the SC correlation (see Table 1), i.e., λs = λsc, which is consistent with the striped PDW. However, our results suggest that the long-range correlation Fy(r) is special to three-leg cylinder. On the contrary, Fy(r) decays exponentially on the wider four-leg cylinders (Fig. 4b), which is similar with both Fx(r) and Fz(r). However, the evidences of the striped PDW are robust for both three and four-leg cylinders with similar sign-changing SC correlations (see Supplementary Discussion for details).

Fig. 4: Spin–spin correlations.
figure 4

a Fz(r) for three-leg cylinder at different doping levels δ and b Fy(r) for both three and four-leg cylinders at doping level δ = 1/12. Inset shows the SDW spatial oscillation, i.e., Fy(r), with the black square and blue triangle symbols are for δ = 1/9 and δ = 1/12, respectively.

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In addition to superconductivity, we have also measured the single-particle Green function \({G}_{\sigma }(r)=\langle {c}_{i}^{\dagger }{c}_{i+r}\rangle\) to test the possibility of Fermi liquid state45. It is clear that the single-particle Green function decays exponentially at long distances as \({G}_{\sigma }(r) \sim {{\rm{e}}}^{-r/{\xi }_{{\rm{G}}}}\) (see Supplementary Discussion for details). Here, ξG is the correlation length, which is summarized in Table 1. As a result, our study suggests that the Fermi liquid state is unlikely in the lightly hole-doped KSL.

Central charge

Our results suggest that the ground state of the lightly hole-doped KSL has quasi-long-range CDW correlation with gapless charge mode. To show this, we have calculated the von Neumann entanglement entropy \(S(x)=-{\rm{Tr}}\left[{\rho }_{x}{\rm{ln}}{\rho }_{x}\right]\), where ρx is the reduced density matrix of subsystem with length x. For finite 1D critical system of length Lx with open boundaries, the central charge c or the number of gapless mode can be obtained using51,52,

$$\begin{array}{lll}S(x)=\frac{c}{6}{\rm{ln}}[\frac{4({L}_{x}+1)}{\pi}\sin\frac{\pi(2x+1)}{2({L}_{x}+1)}| \sin{k}_{{\rm{F}}}|]\\\qquad\quad\;+\;\frac{a\pi\sin[{k}_{{\rm{F}}}(2x+1)]}{4({L}_{x}+1)\sin\frac{\pi(2x+1)}{2({L}_{x}+1)}| \sin{k}_{{\rm{F}}}|}+\tilde{c}.\end{array}$$
(3)

Here, kF denotes the Fermi momentum, a and \(\tilde{c}\) are model-dependent parameters. Figure 5a shows an example of S(x) for three-leg cylinder at doping level δ = 1/9. Here we have calculated S(x) by dividing the system into two parts with smooth boundary53 through both x and y bonds as shown in the inset of Fig. 5b. The extracted central charge c is given in Fig. 5 as a function of Lx at doping levels δ = 1/9 and 1/12. It is clear that the central charge quickly converges to c = 1 with the increase of Lx, although it deviates notably from c = 1 on short cylinders due to finite-size effect. Therefore, our results show that there is one gapless charge mode that is consistent with the quasi-long-range CDW correlations. We have obtained similar results on four-leg cylinders, which are provided in the Supplementary Discussion.

Fig. 5: Central charge.
figure 5

The extracted central charge c as a function of Lx using Eq. (3) for three-leg cylinder at doping level a δ = 1/9 and bδ = 1/12. Error bars denote numerical uncertainty. Inset in a is the entanglement entropy S(x) on three-leg cylinder of length Lx = 48, a few data points in red are removed to minimize the boundary effect. Inset in b shows the smooth boundaries parallel to the e2 direction through x and y-bond labeled by the dashed lines.

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Discussion

Admittedly, the DMRG calculations are carried out on finite length cylinders. However, based on the results we have obtained we conjecture that the exact ground state for an infinite long three-leg cylinder has the following properties: (1) There is a single gapless charge mode and a gap (which produces exponentially falling correlations) for all spin carrying excitations. (2) Long-range SDW order with the ordered moment oriented in the y direction - that is along the circumference of the cylinder—and an ordering vector Q = 3πδ/2; the connected spin correlations fall exponentially with a correlation length ξs ~ a0. (3) There are power-law CDW correlations with an exponent Kc ~ 2/3 and an ordering vector K = 2Q. (4) There are strong even-parity d-wave-like PDW correlations with an ordering vector Q, which fall either exponentially with a correlation length ξsc ~ 3a0 or possibly with a high power-law Ksc ≥ 4. 5) All other forms of SC correlations—those corresponding to odd-parity pairing or spatially uniform even-parity pairing—are much weaker in comparison with the PDW correlations.

There are many aspects of these observations that are surprising, and will need to be understood theoretically. The PDW correlations are sufficiently short-ranged that one would infer (based on any reasonable conjecture concerning their time dependence) that the corresponding PDW susceptibility would be finite. Thus, at present, we can only conclude that the present results are suggestive of a possible PDW ordered state for the 2D (infinite leg) version of this model. However, it is notable that the favored forms of order are remarkably reminiscent of those conjectured to be present in the cuprate high-temperature superconductor, LBCO, where direct evidence exists of stripe SDW order with wavevector Q ≈ 2πδ and CDW order with ordering vector K = 2Q, and indirect evidence has been adduced for PDW order with ordering vector Q.

In this paper, we primarily focus on the lightly doped Kitaev model, it will be interesting to study the higher doping case as well as the extend Kitaev model with further neighbor hopping, which is shown to be essential to enhance the superconductivity on the square lattice19,20. As other terms such as the Heisenberg interaction and spin-orbit couplings are also present in real materials such as Na2IrO3 and α-RuCl335,36,37,38,39,40, it will be interesting to study these systems as well.

Methods

Model Hamiltonian

We employ DMRG46 to investigate the ground state properties of the hole-doped Kitaev model on the honeycomb lattice defined by the Hamiltonian

$$H=-t\mathop{\sum}\limits_{\langle ij\rangle ,\sigma }({c}_{i\sigma }^{\dagger }{c}_{j\sigma }+h.c.)+{J}_{{\rm{K}}}\mathop{\sum}\limits_{\langle ij\rangle }{S}_{i}^{\gamma }{S}_{j}^{\gamma }.$$
(4)

Here, \({c}_{i\sigma }^{\dagger }\)(ciσ) is the electron creation (annihilation) operator with spin-σ on site i = (xi, yi), \({S}_{i}^{\gamma }\) is the γ-component of the S = 1/2 spin operator on site i, where γ = x, y, z labels the three different links of the hexagonal lattice as illustrated in Fig. 1. 〈ij〉 denotes nearest-neighbor (NN) sites and the Hilbert space is constrained by the no-double occupancy condition, ni ≤ 1, where \({n}_{i}={\sum }_{\sigma }{c}_{i\sigma }^{\dagger }{c}_{i\sigma }\) is the electron number operator. At half-filling, i.e., ni = 1, Eq. (4) reduces to the Kitaev model, which is known to have a gapless spin liquid ground state that can be gapped out into a non-Abelian topological phase by certain time-reversal symmetry perturbations34,54.

Numerical details

The lattice geometry used in our simulations is depicted in Fig. 1, where \({{\bf{e}}}_{1}=(\sqrt{3},0)\) and \({{\bf{e}}}_{2}=(\sqrt{3}/2,3/2)\) denote the two basis vectors. We consider honeycomb cylinders with periodic and open boundary conditions in the e2 and e1 directions, respectively. We focus on cylinders of width Ly and length Lx, where Ly and Lx are the number of unit cells (Ly and \({\tilde{L}}_{x}=2{L}_{x}\) are the number of sites) along the e2 and e1 directions, respectively. The total number of sites is N = 2 × Ly × Lx. The hole doping concentration is defined as δ = Nh/N, where Nh is the number of doped holes. We set JK = 1 as an energy unit and consider t = 3. In this paper, we focus primarily on three-leg cylinders, i.e., Ly = 3a0, of length Lx = 12a0 ~ 48a0 (i.e., \({\tilde{L}}_{x}=24 \sim 96\)), at doping levels δ = 1/12 and 1/9. We keep up to m = 8000 number of states in each DMRG block with a typical truncation error ϵ ~ 10−6. We find similar results on four-leg cylinders. Further details of the numerical simulation are provided in the Supplementary Discussion.