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arXiv:1912.06624 · npj Quantum Materials

Superconductivity in the doped quantum spin liquid on the triangular lattice

Hong‐Chen Jiang

Published Aug 6, 2021• 32 citations
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Abstract

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.

DOI
10.1038/s41535-021-00375-w

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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, χsc ∼ T−(2 − Ksc){\chi }_{sc} \,\sim\, {T}^{-(2\,-\,{K}_{sc})}χsc​∼T−(2−Ksc​) 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.
We have performed density-matrix renormalization group studies of a square lattice $t\text{\ensuremath{-}}J$ model with small hole doping, $\ensuremath{\delta}\ensuremath{\ll}1$, on long four and six-leg cylinders. We include frustration in the form of a second-neighbor exchange coupling, J2=J1/2{J}_{2}={J}_{1}/2J2​=J1​/2, such that the undoped ($\ensuremath{\delta}=0$) ``parent'' state is a quantum spin liquid. In contrast to the relatively short range superconducting (SC) correlations that have been observed in recent studies of the six-leg cylinder in the absence of frustration, we find power-law SC correlations with a Luttinger exponent, ${K}_{\mathrm{SC}}\ensuremath{\approx}1$, consistent with a strongly diverging SC susceptibility, $\ensuremath{\chi}\ensuremath{\sim}{T}^{\ensuremath{-}(2\ensuremath{-}{K}_{\mathrm{SC}})}$ as the temperature $T\ensuremath{\rightarrow}0$. The spin-spin correlations---as in the undoped state---fall exponentially suggesting that the SC ``pairing'' correlations evolve smoothly from the insulating parent state.
It has long been proposed that doping a chiral spin liquid (CSL) or fractional quantum Hall state can give rise to topological superconductivity. Despite intensive effort, definitive evidences still remain lacking. We address this problem by studying the t-J model supplemented by time-reversal symmetry breaking chiral interaction J_{χ} on the triangular lattice using density-matrix renormalization group with a finite concentration δ of doped holes. It has been established that the undoped, i.e., δ=0, system has a CSL ground state in the parameter region 0.32≤J_{χ}/J≤0.56. Upon light doping, we find that the ground state of the system is consistent with a Luther-Emery liquid with power-law superconducting and charge-density-wave correlations but short-range spin-spin correlations. In particular, the superconducting correlations, whose pairing symmetry is consistent with d±id wave, are dominant at all hole doping concentrations. Our results provide direct evidences that doping the CSL on the triangular lattice can naturally give rise to topological superconductivity.
Abstract The authors review recent density‐matrix renormalization group studies of lightly doped quantum spin liquids (QSLs) on the kagome lattice. While a number of distinct conducting phases, including high temperature superconductivity, have been theoretically anticipated, a tendency toward fractionalized insulating charge‐density‐wave (CDW) states is instead found. In agreement with earlier work [Jiang, Devereaux, and Kivelson, Phys. Rev. Lett . 2017, 119 , 067002], results for the t ‐ J model reveal that, starting from a fully gapped QSL, light doping leads to CDW long‐range order with a pattern that depends on lattice geometry and doping concentration such that there is one doped‐hole per CDW unit cell, while the spin–spin correlations remain short‐ranged. Alternatively, this state can be viewed as a stripe crystal or Wigner crystal of spinless holons, rather than of doped holes. From here, by studying generalized versions of the t ‐ J model, these results are extended to light doping of other types of QSLs, including critical and chiral QSLs. These results suggest that doping these QSLs also leads to insulating states with long‐range CDW order. While the superconducting correlations are short‐ranged, they can be significantly enhanced by second‐neighbor electron hopping. The relevance of these numerical results to Kagome materials is also discussed.
We address the problem of a lightly doped spin liquid through a large-scale density-matrix renormalization group study of the t-J model on a kagome lattice with a small but nonzero concentration δ of doped holes. It is now widely accepted that the undoped (δ=0) spin-1/2 Heisenberg antiferromagnet has a spin-liquid ground state. Theoretical arguments have been presented that light doping of such a spin liquid could give rise to a high temperature superconductor or an exotic topological Fermi liquid metal. Instead, we infer that the doped holes form an insulating charge-density wave state with one doped hole per unit cell, i.e., a Wigner crystal. Spin correlations remain short ranged, as in the spin-liquid parent state, from which we infer that the state is a crystal of spinless holons, rather than of holes. Our results may be relevant to kagome lattice herbertsmithite upon doping.
Superconductivity in doped quantum paramagnets has been a subject of long theoretical inquiry. In this work, we report a density matrix renormalization group study of lightly doped $t\text{\ensuremath{-}}J$ models on the finite-width square lattice (doped hole densities $\ensuremath{\delta}=1/12$ and 1/8) with parameters for which previous studies have suggested that the undoped system in 2D is either a quantum spin liquid or a valence bond crystal. Our studies are performed on cylinders with width up to 8. Ground-state correlations are found to be nearly identical for the ``doped quantum spin liquid'' and ``doped valence bond crystal.'' Upon increasing the cylinder widths from 4 to 8, we observed a significant strengthening of the quasi-long-range superconducting correlations and a dramatic suppression of any ``competing'' charge density wave order. Extrapolating from the observed behavior of the width eight cylinders, we speculate that the system has a nodeless ddd-wave superconducting ground state in the 2D limit.
It has long been believed that doped quantum spin liquids (QSLs) can give rise to fascinating quantum phases, including the possibility of high-temperature superconductivity (SC) as proposed by P. W. Anderson's resonating valence bond (RVB) scenario. The Kagome lattice ttt-JJJ model is known to exhibit spin liquid behavior at half-filling, making it an ideal system for studying the properties of doped QSL. In this study, we employ the fermionic projected entangled simplex state (PESS) method to investigate the ground state properties of the Kagome lattice ttt-JJJ model with t/J=3.0t/J = 3.0t/J=3.0. Our results reveal a phase transition from charge density wave (CDW) states to uniform states around a critical doping level δc≈0.27δ_c \approx 0.27δc​≈0.27. Within the CDW phase, we observe different types of Wigner crystal (WC) formulated by doped holes that are energetically favored. As we enter the uniform phase, a non-Fermi liquid (NFL) state emerges within the doping range 0.27<δ<0.320.27 < δ< 0.320.27<δ<0.32, characterized by an exponential decay of all correlation functions. With further hole doping, we discover the appearance of a pair density wave (PDW) state within a narrow doping region 0.32<δ<1/30.32 < δ< 1/30.32<δ<1/3. We also discuss the potential experimental implications of our findings.