arXiv:2210.06847 · Physical review. B./Physical review. B

Superconductivity in doped triangular Mott insulators: The roles of parent spin backgrounds and charge kinetic energy

Zheng Zhu, Qianqian Chen

Published Jun 6, 2023• 8 citations

Abstract

We study the prerequisites for realizing superconductivity in doped triangular-lattice Mott insulators by considering three distinct parent spin backgrounds, i.e., ${120}^{\ensuremath{\circ}}$ antiferromagnets, quantum spin liquid, and stripy antiferromagnets, and all possible sign combinations $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$ of nearest-neighbor hopping and next-nearest-neighbor hopping $({t}_{1},{t}_{2})$. Based on density matrix renormalization group calculations, we find that, with finite ${t}_{2}$ and specific sign combinations $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$, the quasi-long-range superconductivity order can always be achieved, regardless of the nature of the parent spin backgrounds. Besides specific hopping signs $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$, these superconductivity phases in triangular lattices are commonly characterized by short-ranged spin correlations and two charges per stripe. In the robust superconductivity phase realized at larger ${t}_{2}/{t}_{1}$, flipping the signs ${\ensuremath{\tau}}_{2}$ and ${\ensuremath{\tau}}_{1}$ gives rise to the stripe phase without strong pairing and a pseudogaplike phase without Cooper-pair phase coherence, respectively. Interestingly, the roles of the two hopping signs are switched at smaller ${t}_{2}/{t}_{1}$. Moreover, different sign combinations $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$ would stabilize distinct phases including superconductivity, charge density waves, spin density waves, and pseudogaplike phases accordingly. Our findings suggest the important role of charge kinetic energy in realizing superconductivity in doped triangular-lattice Mott insulators.

DOI: 10.1103/physrevb.107.l220502

Related papers

Robust d -Wave Superconductivity in the Square-Lattice t − J Model

Shou-Shu Gong, Wei Kang Zhu, D. N. Sheng

arXiv:2104.03758Physical Review LettersAug 27, 202187 citations

Sign structure of the t − t ′ − J model and its physical consequences

Xin Lu, Jiaxin Zhang, Shou-Shu Gong +2 more

arXiv:2303.13498Physical review. B./Physical review. BOct 10, 202415 citations

Superconductivity in the doped quantum spin liquid on the triangular lattice

Hong‐Chen Jiang

arXiv:1912.06624npj Quantum MaterialsAug 6, 202132 citations

Searching for superconductivity in doped triangular lattice Kitaev magnets

Andrew Hardy, Ryan Levy, Arun Paramekanti

arXiv:2508.16720Physical review. B./Physical review. BApr 22, 20262 citations

Doped Mott insulators in the triangular-lattice Hubbard model

Zheng Zhu, D. N. Sheng, Ashvin Vishwanath

arXiv:2007.11963Physical review. B./Physical review. BMay 10, 202257 citations

Superconductivity and competing orders in honeycomb $t$-$J$ model: interplay of lattice geometry and next-nearest-neighbor hopping

Zhi Xu, Hong‐Chen Jiang, Yi‐Fan Jiang

arXiv:2604.08937Apr 10, 20260 citations

arXiv:2210.06847 · Physical review. B./Physical review. B

Superconductivity in doped triangular Mott insulators: The roles of parent spin backgrounds and charge kinetic energy

Zheng Zhu, Qianqian Chen

Published Jun 6, 2023• 8 citations

arXiv HTML PDF

Abstract

We study the prerequisites for realizing superconductivity in doped triangular-lattice Mott insulators by considering three distinct parent spin backgrounds, i.e.,

{120}^{\ensuremath{\circ}}

antiferromagnets, quantum spin liquid, and stripy antiferromagnets, and all possible sign combinations

({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})

of nearest-neighbor hopping and next-nearest-neighbor hopping

({t}_{1},{t}_{2})

. Based on density matrix renormalization group calculations, we find that, with finite

{t}_{2}

and specific sign combinations

({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})

, the quasi-long-range superconductivity order can always be achieved, regardless of the nature of the parent spin backgrounds. Besides specific hopping signs

({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})

, these superconductivity phases in triangular lattices are commonly characterized by short-ranged spin correlations and two charges per stripe. In the robust superconductivity phase realized at larger

{t}_{2}/{t}_{1}

, flipping the signs

{\ensuremath{\tau}}_{2}

and

{\ensuremath{\tau}}_{1}

gives rise to the stripe phase without strong pairing and a pseudogaplike phase without Cooper-pair phase coherence, respectively. Interestingly, the roles of the two hopping signs are switched at smaller

{t}_{2}/{t}_{1}

. Moreover, different sign combinations

({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})

would stabilize distinct phases including superconductivity, charge density waves, spin density waves, and pseudogaplike phases accordingly. Our findings suggest the important role of charge kinetic energy in realizing superconductivity in doped triangular-lattice Mott insulators.

DOI: 10.1103/physrevb.107.l220502

Related papers

Robust d -Wave Superconductivity in the Square-Lattice t − J Model

Shou-Shu Gong, Wei Kang Zhu, D. N. Sheng

Unravelling competing orders emergent in doped Mott insulators and their interplay with unconventional superconductivity is one of the major challenges in condensed matter physics. To explore the possible superconducting state in a doped Mott insulator, we study the square-lattice t-J model with both the nearest-neighbor and next-nearest-neighbor electron hoppings and spin interactions. By using the state-of-the-art density matrix renormalization group calculation with imposing charge U(1) and spin SU(2) symmetries on the six-leg cylinders, we establish a quantum phase diagram including three phases: a stripe charge density wave phase, a superconducting phase without static charge order, and a superconducting phase coexistent with a weak charge stripe order. Crucially, we demonstrate that the superconducting phase has a power-law pairing correlation that decays much slower than the charge density and spin correlations, which is a quasi-1D descendant of the uniform d-wave superconductor in two dimensions. These findings reveal that enhanced charge and spin fluctuations with optimal doping is able to produce robust d-wave superconductivity in doped Mott insulators, providing a foundation for connecting theories of superconductivity to models of strongly correlated systems.Unravelling competing orders emergent in doped Mott insulators and their interplay with unconventional superconductivity is one of the major challenges in condensed matter physics. To explore the possible superconducting state in a doped Mott insulator, we study the square-lattice t-J model with both the nearest-neighbor and next-nearest-neighbor electron hoppings and spin interactions. By using the state-of-the-art density matrix renormalization group calculation with imposing charge U(1) and spin SU(2) symmetries on the six-leg cylinders, we establish a quantum phase diagram including three phases: a stripe charge density wave phase, a superconducting phase without static charge order, and a superconducting phase coexistent with a weak charge stripe order. Crucially, we demonstrate that the superconducting phase has a power-law pairing correlation that decays much slower than the charge density and spin correlations, which is a quasi-1D descendant of the uniform d-wave superconductor in two dimensions. These findings reveal that enhanced charge and spin fluctuations with optimal doping is able to produce robust d-wave superconductivity in doped Mott insulators, providing a foundation for connecting theories of superconductivity to models of strongly correlated systems.

arXiv:2104.03758Physical Review LettersAug 27, 202187 citations

Sign structure of the t − t ′ − J model and its physical consequences

Xin Lu, Jiaxin Zhang, Shou-Shu Gong +2 more

Understanding the doped Mott insulator is a central challenge in condensed matter physics. In this work, we first explicitly identify a new sign structure in the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model on the square lattice that replaces the conventional Fermi statistics for weakly interacting electrons. Then we show that the singular, i.e., the phase-string part of the sign structure in the partition function can be precisely turned off in a modified model. The density matrix renormalization group method is then employed to study these two models comparatively on finite-size systems, which is designed to unveil the consequences of the phase-string component. We find that the hole pairing is present not only in the quasi-long-range superconducting phase but also in the stripe phase of the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model. However, once the phase-string is switched off, both the superconducting and stripe orders together with the underlying hole pairing disappear. The corresponding ground state reduces to a trivial Fermi-liquid-like state with small hole Fermi pockets that is decoupled from the antiferromagnetic spin background. It is in sharp contrast to the original $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model where large Fermi surfaces can be restored in the stripe phase found at ${t}^{\ensuremath{'}}/t<0$ or the superconducting phase at ${t}^{\ensuremath{'}}/t>0$ in the six-leg ladder calculation. Our study clearly demonstrates that the strong correlation effect in doped Mott insulator should be mainly attributed to the long-range quantum entanglement between the spin and charge, which is, nonperturbatively, beyond a simple spin-charge separation under the no double occupancy constraint.Understanding the doped Mott insulator is a central challenge in condensed matter physics. In this work, we first explicitly identify a new sign structure in the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model on the square lattice that replaces the conventional Fermi statistics for weakly interacting electrons. Then we show that the singular, i.e., the phase-string part of the sign structure in the partition function can be precisely turned off in a modified model. The density matrix renormalization group method is then employed to study these two models comparatively on finite-size systems, which is designed to unveil the consequences of the phase-string component. We find that the hole pairing is present not only in the quasi-long-range superconducting phase but also in the stripe phase of the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model. However, once the phase-string is switched off, both the superconducting and stripe orders together with the underlying hole pairing disappear. The corresponding ground state reduces to a trivial Fermi-liquid-like state with small hole Fermi pockets that is decoupled from the antiferromagnetic spin background. It is in sharp contrast to the original $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model where large Fermi surfaces can be restored in the stripe phase found at ${t}^{\ensuremath{'}}/t<0$ or the superconducting phase at ${t}^{\ensuremath{'}}/t>0$ in the six-leg ladder calculation. Our study clearly demonstrates that the strong correlation effect in doped Mott insulator should be mainly attributed to the long-range quantum entanglement between the spin and charge, which is, nonperturbatively, beyond a simple spin-charge separation under the no double occupancy constraint.

arXiv:2303.13498Physical review. B./Physical review. BOct 10, 202415 citations

Superconductivity in the doped quantum spin liquid on the triangular lattice

Hong‐Chen Jiang

Abstract Broad interest in quantum spin liquid (QSL) phases was triggered by the notion that they can be viewed as insulating phases with preexisting electron pairs, such that upon light doping they might automatically yield high temperature superconductivity. Yet despite intense experimental and numerical efforts, definitive evidence showing that doping QSLs leads to superconductivity has been lacking. We address the problem of a lightly doped QSL through a large-scale density-matrix renormalization group study of the t-J model on finite-circumference triangular cylinders with a small but nonzero concentration of doped holes. We provide direct evidences that doping QSL can naturally give rise to d-wave superconductivity. Specifically, we find power-law superconducting correlations with a Luttinger exponent, Ksc ≈ 1, which is consistent with a strongly diverging superconducting susceptibility, ${\chi }_{sc} \,\sim\, {T}^{-(2\,-\,{K}_{sc})}$ χ s c ~ T − ( 2 − K s c ) as the temperature T → 0. The spin–spin correlations—as in the undoped QSL state—fall exponentially which suggests that the superconducting pair-pair correlations evolve smoothly from the insulating parent state.Abstract Broad interest in quantum spin liquid (QSL) phases was triggered by the notion that they can be viewed as insulating phases with preexisting electron pairs, such that upon light doping they might automatically yield high temperature superconductivity. Yet despite intense experimental and numerical efforts, definitive evidence showing that doping QSLs leads to superconductivity has been lacking. We address the problem of a lightly doped QSL through a large-scale density-matrix renormalization group study of the t-J model on finite-circumference triangular cylinders with a small but nonzero concentration of doped holes. We provide direct evidences that doping QSL can naturally give rise to d-wave superconductivity. Specifically, we find power-law superconducting correlations with a Luttinger exponent, Ksc ≈ 1, which is consistent with a strongly diverging superconducting susceptibility, $${\chi }_{sc} \,\sim\, {T}^{-(2\,-\,{K}_{sc})}$$ χ s c ~ T − ( 2 − K s c ) as the temperature T → 0. The spin–spin correlations—as in the undoped QSL state—fall exponentially which suggests that the superconducting pair-pair correlations evolve smoothly from the insulating parent state.

arXiv:1912.06624npj Quantum MaterialsAug 6, 202132 citations

Searching for superconductivity in doped triangular lattice Kitaev magnets

Andrew Hardy, Ryan Levy, Arun Paramekanti

Motivated by exploring correlated metals with frustrating bond-dependent exchange interactions, we study hole and electron doped Kitaev Mott insulators on the triangular lattice. Using homogeneous parton mean field theory, we find that the stripe antiferromagnetic (AFM) order for Kitaev coupling $K>0$ and the ferromagnetic (FM) order for $K<0$ , both vanish at sufficiently large doping, beyond which we find regimes with chiral $d\pm i d$ singlet pairing and $p\pm ip$ triplet pairing respectively. Our tensor network computations however reveal that the superconducting correlations are strongly suppressed; while FM order stubbornly persists for the doped $K<0$ model, the doped $K>0$ model features emergent spin-charge modulated stripe orders. At higher hole doping for $K > 0$ , where AFM order is more strongly suppressed than for the electron doped case, incorporating a sufficiently strong nearest-neighbor attraction yields evidence for singlet $d$ -wave superconductivity with Luttinger parameter $K_{\rm sc} < 1$ . Our work sets the stage for a broader exploration of doping effects in triangular lattice magnets such as NaRuO $_2$ which feature bond-dependent exchange interactions.Motivated by exploring correlated metals with frustrating bond-dependent exchange interactions, we study hole and electron doped Kitaev Mott insulators on the triangular lattice. Using homogeneous parton mean field theory, we find that the stripe antiferromagnetic (AFM) order for Kitaev coupling $K>0$ and the ferromagnetic (FM) order for $K<0$, both vanish at sufficiently large doping, beyond which we find regimes with chiral $d\pm i d$ singlet pairing and $p\pm ip$ triplet pairing respectively. Our tensor network computations however reveal that the superconducting correlations are strongly suppressed; while FM order stubbornly persists for the doped $K<0$ model, the doped $K>0$ model features emergent spin-charge modulated stripe orders. At higher hole doping for $K > 0$, where AFM order is more strongly suppressed than for the electron doped case, incorporating a sufficiently strong nearest-neighbor attraction yields evidence for singlet $d$-wave superconductivity with Luttinger parameter $K_{\rm sc} < 1$. Our work sets the stage for a broader exploration of doping effects in triangular lattice magnets such as NaRuO$_2$ which feature bond-dependent exchange interactions.

arXiv:2508.16720Physical review. B./Physical review. BApr 22, 20262 citations

Doped Mott insulators in the triangular-lattice Hubbard model

Zheng Zhu, D. N. Sheng, Ashvin Vishwanath

We investigate the evolution of the Mott insulators in the triangular lattice Hubbard Model, as a function of hole doping $\ensuremath{\delta}$ in both the strong and intermediate coupling limits. Using the advanced density matrix renormalization group (DMRG) method, at light hole doping $\ensuremath{\delta}\ensuremath{\lesssim}10%$, we find a significant difference between strong and intermediate couplings. Notably, at intermediate coupling an unusual metallic state emerges, with short ranged spin correlations but long ranged spin-chirality order. Moreover, no clear Fermi surface or wave vector is observed, this chiral metal also exhibits staggered loop current, which breaks the translational symmetry. These features disappear on increasing interaction strength or on further doping. At strong coupling, the 120 degree magnetic order of the insulating magnet persists for light doping, and produces hole pockets with a well-defined Fermi surface. On further doping, $\ensuremath{\delta}\ensuremath{\approx}10%\ensuremath{\sim}20%$ SDW order and coherent hole Fermi pockets are found at both strong and intermediate couplings. At even higher doping $\ensuremath{\delta}\ensuremath{\gtrsim}20%$, the SDW order is suppressed and the spin-singlet Cooper pair correlations are simultaneously enhanced. We also briefly comment on the strong particle-hole asymmetry of the model.We investigate the evolution of the Mott insulators in the triangular lattice Hubbard Model, as a function of hole doping $\ensuremath{\delta}$ in both the strong and intermediate coupling limits. Using the advanced density matrix renormalization group (DMRG) method, at light hole doping $\ensuremath{\delta}\ensuremath{\lesssim}10%$, we find a significant difference between strong and intermediate couplings. Notably, at intermediate coupling an unusual metallic state emerges, with short ranged spin correlations but long ranged spin-chirality order. Moreover, no clear Fermi surface or wave vector is observed, this chiral metal also exhibits staggered loop current, which breaks the translational symmetry. These features disappear on increasing interaction strength or on further doping. At strong coupling, the 120 degree magnetic order of the insulating magnet persists for light doping, and produces hole pockets with a well-defined Fermi surface. On further doping, $\ensuremath{\delta}\ensuremath{\approx}10%\ensuremath{\sim}20%$ SDW order and coherent hole Fermi pockets are found at both strong and intermediate couplings. At even higher doping $\ensuremath{\delta}\ensuremath{\gtrsim}20%$, the SDW order is suppressed and the spin-singlet Cooper pair correlations are simultaneously enhanced. We also briefly comment on the strong particle-hole asymmetry of the model.

arXiv:2007.11963Physical review. B./Physical review. BMay 10, 202257 citations

Superconductivity and competing orders in honeycomb $t$ - $J$ model: interplay of lattice geometry and next-nearest-neighbor hoppingSuperconductivity and competing orders in honeycomb $t$-$J$ model: interplay of lattice geometry and next-nearest-neighbor hopping

Zhi Xu, Hong‐Chen Jiang, Yi‐Fan Jiang

We investigate the extended $t$ - $J$ model on honeycomb lattices with next-nearest-neighbor (NNN) electron hopping $t'$ and superexchange coupling $J'=(t'/t)^2 J$ using large-scale density-matrix renormalization group (DMRG) simulations and slave-boson mean-field theory (SBMFT). By systematically varying $t'$ and cylinder geometries, our DMRG results reveal several competing phases with distinct charge and superconducting (SC) properties. On YC4-0 cylinders possessing bonds lying along $\vec{e}_y$ direction, the ground state of doped models exhibits pronounced quasi-long-range $d$ -wave SC with coexisting armchair-oriented stripes (a-stripe) across a broad range of $t'$ . Notably, the SC Luttinger exponent has a non-monotonic dependence on $t'$ , showing an optimal $t'_{op}\sim0.4$ for dominant SC. Conversely, XC cylinders host a competing long-range zigzag stripes phase without SC for $t'>0.5$ , highlighting the role of boundary geometry in stabilizing distinct competing phases in DMRG. To elucidate the stability of all these competing phases in 2D limit, we employ SBMFT and identify the a-stripe as the stable configuration across most of phase diagram, with a transition to uniform nematic $d$ -wave SC at large $t'$ for $δ=1/8$ . The combined results from two complementary approaches suggest a robust $t'$ -induced SC phase that might remain stable in doped extended $t$ - $J$ model on the honeycomb lattice.We investigate the extended $t$-$J$ model on honeycomb lattices with next-nearest-neighbor (NNN) electron hopping $t'$ and superexchange coupling $J'=(t'/t)^2 J$ using large-scale density-matrix renormalization group (DMRG) simulations and slave-boson mean-field theory (SBMFT). By systematically varying $t'$ and cylinder geometries, our DMRG results reveal several competing phases with distinct charge and superconducting (SC) properties. On YC4-0 cylinders possessing bonds lying along $\vec{e}_y$ direction, the ground state of doped models exhibits pronounced quasi-long-range $d$-wave SC with coexisting armchair-oriented stripes (a-stripe) across a broad range of $t'$. Notably, the SC Luttinger exponent has a non-monotonic dependence on $t'$, showing an optimal $t'_{op}\sim0.4$ for dominant SC. Conversely, XC cylinders host a competing long-range zigzag stripes phase without SC for $t'>0.5$, highlighting the role of boundary geometry in stabilizing distinct competing phases in DMRG. To elucidate the stability of all these competing phases in 2D limit, we employ SBMFT and identify the a-stripe as the stable configuration across most of phase diagram, with a transition to uniform nematic $d$-wave SC at large $t'$ for $δ=1/8$. The combined results from two complementary approaches suggest a robust $t'$-induced SC phase that might remain stable in doped extended $t$-$J$ model on the honeycomb lattice.

arXiv:2604.08937Apr 10, 20260 citations

We study the prerequisites for realizing superconductivity in doped triangular-lattice Mott insulators by considering three distinct parent spin backgrounds, i.e.,

{120}^{\ensuremath{\circ}}

antiferromagnets, quantum spin liquid, and stripy antiferromagnets, and all possible sign combinations

({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})

of nearest-neighbor hopping and next-nearest-neighbor hopping

({t}_{1},{t}_{2})

. Based on density matrix renormalization group calculations, we find that, with finite

{t}_{2}

and specific sign combinations

({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})

, the quasi-long-range superconductivity order can always be achieved, regardless of the nature of the parent spin backgrounds. Besides specific hopping signs

({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})

{t}_{2}/{t}_{1}

, flipping the signs

{\ensuremath{\tau}}_{2}

and

{\ensuremath{\tau}}_{1}

{t}_{2}/{t}_{1}

. Moreover, different sign combinations

({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})

{\chi }_{sc} \,\sim\, {T}^{-(2\,-\,{K}_{sc})}

χ s c ~ T − ( 2 − K s c ) as the temperature T → 0. The spin–spin correlations—as in the undoped QSL state—fall exponentially which suggests that the superconducting pair-pair correlations evolve smoothly from the insulating parent state.

K>0

and the ferromagnetic (FM) order for

K<0

, both vanish at sufficiently large doping, beyond which we find regimes with chiral

d\pm i d

singlet pairing and

p\pm ip

triplet pairing respectively. Our tensor network computations however reveal that the superconducting correlations are strongly suppressed; while FM order stubbornly persists for the doped

K<0

model, the doped

K>0

model features emergent spin-charge modulated stripe orders. At higher hole doping for

K > 0

, where AFM order is more strongly suppressed than for the electron doped case, incorporating a sufficiently strong nearest-neighbor attraction yields evidence for singlet

d

-wave superconductivity with Luttinger parameter

K_{\rm sc} < 1

. Our work sets the stage for a broader exploration of doping effects in triangular lattice magnets such as NaRuO

_2

which feature bond-dependent exchange interactions.

Superconductivity and competing orders in honeycomb

t

J

model: interplay of lattice geometry and next-nearest-neighbor hopping

We investigate the extended

t

J

model on honeycomb lattices with next-nearest-neighbor (NNN) electron hopping

t'

and superexchange coupling

J'=(t'/t)^2 J

using large-scale density-matrix renormalization group (DMRG) simulations and slave-boson mean-field theory (SBMFT). By systematically varying

t'

and cylinder geometries, our DMRG results reveal several competing phases with distinct charge and superconducting (SC) properties. On YC4-0 cylinders possessing bonds lying along

\vec{e}_y

direction, the ground state of doped models exhibits pronounced quasi-long-range

d

-wave SC with coexisting armchair-oriented stripes (a-stripe) across a broad range of

t'

. Notably, the SC Luttinger exponent has a non-monotonic dependence on

t'

, showing an optimal

t'_{op}\sim0.4

for dominant SC. Conversely, XC cylinders host a competing long-range zigzag stripes phase without SC for

t'>0.5

, highlighting the role of boundary geometry in stabilizing distinct competing phases in DMRG. To elucidate the stability of all these competing phases in 2D limit, we employ SBMFT and identify the a-stripe as the stable configuration across most of phase diagram, with a transition to uniform nematic

d

-wave SC at large

t'

for

δ=1/8

. The combined results from two complementary approaches suggest a robust

t'

-induced SC phase that might remain stable in doped extended

t

J

model on the honeycomb lattice.