The behavior of thermal fluctuations of a superconducting order parameter with extended $ s $ and chiral $ d + id $ symmetry is investigated. The study is carried out on a triangular lattice within the framework of the quasi-two-dimensional single-band model with attraction between electrons at neighboring sites. The method of consistent consideration of the order parameter fluctuations and the charge carrier scattering by fluctuations of coupled electron pairs, based on the theory of functional integration is used. The distribution functions of the phase fluctuation probabilities depending on temperature and charge carrier concentration are obtained. The temperature dependences of the amplitudes of the averaged superconducting order parameter are calculated. A phase diagram of superconducting states is constructed for the entire range of variation in the charge carrier concentration $ 0 <n <2 $. Near the boundaries of this range, topologically trivial superconducting states with extended $ s $ symmetry are realized, while a superconducting state with topologically nontrivial chiral $ d + id $ symmetry is realized between them. The calculated anomalous self-energies are compared with the experimental ones obtained using machine learning techniques.
Superconducting Pairing Symmetries in Anisotropic Triangular Quantum Antiferromagnets
最近在层状有机化合物$κ$ -（ET）$_2$ Cu $_2$（CN）$_3$中发现了一种低温自旋液相，在压力下可以变成超导体。基于这一发现，我们研究了各向异性三角晶格上Hubbard-Heisenberg模型中Mott绝缘态和超导态的相变。我们用重整化平均场论研究了Gutzwiller投影BCS波函数。半充满电子系统在大的在位排斥$U$下是莫特绝缘体，在中等的$U$下是超导体。SC态的对称性依赖于各向异性，在各向同性极限附近与$d_{x^2-y^2}+id_{xy}$对称性有隙，在小的各向异性比下与$d_{x^2-y^2}$对称性无隙.
It has long been proposed that doping a chiral spin liquid (CSL) or fractional quantum Hall state can give rise to topological superconductivity. Despite of 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 $0.32\le J_χ/J \le 0.56$. Upo…
Single domain magnetic helicity and triangular chirality in structurally enantiopure Ba3NbFe3Si2O14
在结构手性硅酸镓化合物Ba $_3$ NbFe $_3$ Si $_2$ O $_{14}$中发现了一种新的双手性磁序.磁矩分布在三角形单元的平面阻挫三角形晶格上。在每一个上面，它们形成相同的三角形结构。这种铁手性排列是从平面到平面螺旋调制的。非极化中子散射单晶与球形中子极化证明，一个单一的三角形手性连同一个单一的螺旋是稳定的对映体纯晶体。平均场分析允许辨别在平面到平面超超交换路径中的扭曲的这种选择的相关性。
Frustration-induced insulating chiral spin state in itinerant triangular-lattice magnets
结合精确对角化和Monte-Carlo方法，研究了三角格子上具有竞争超交换作用的半满双交换模型。我们发现，在预期的巡回铁磁和$120^{\circ}$ Yafet-Kittel阶段之间的一个强大的标量手性，绝缘自旋状态出现。在有限温度下，铁磁体标量手征量子临界点的特征是在阻挫巡游磁体R $_2$ Mo $_{2}$ O $_7$中观察到的电荷输运中的异常坏金属行为.
Topological Chiral Superconductivity in the Triangular-Lattice Hofstadter-Hubbard Model
The Hamburg Centre for Ultrafast Imaging and Institute for Quantum Physics, University of Hamburg（汉堡超快成像中心与量子物理研究所，汉堡大学）；Department of Physics and Astronomy, California State University Northridge（加州州立大学北岭分校物理与天文系）；Kavli Institute for Theoretical Physics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校卡弗里理论物理研究所）；French American Center for Theoretical Science, CNRS, KITP, Santa Barbara（法国美国理论科学中心，法国国家科学研究中心，KITP，圣塔芭芭拉）；Canadian Institute for Advanced Research（加拿大高级研究院）
The possible ground states of the undoped and doped Kitaev-Heisenberg model on a triangular lattice are studied. For the undoped system, a combination of the numerical exact diagonalization calculation and the four-sublattice transformation analysis suggests one possible exotic phase and four magnetically ordered phases including a collinear stripe pattern and a noncollinear spiral pattern in the global phase diagram. The exotic phase near the antiferromagnetic (AF) Kitaev point is further investigated by using the Schwinger-fermion mean-field method, and we obtain an energetically favorable $…
Bilayer Moiré structures are a highly tunable laboratory to investigate the physics of strongly correlated electron systems. Moiré transition metal dichalcogenides at low-energies, in particular, are believed to be described by a single narrow band Hubbard model on a triangular lattice with spin-orbit coupling. Motivated by recent experimental evidence for superconductivity in twisted bilayer materials, we investigate the possible superconducting pairings in a two-dimensional single band Rashba-Hubbard model. Using a random-phase approximation in the presence of nearest and next-nearest neighb…
Recent numerical simulations find a possible chiral spin liquid state in the intermediate coupling regime of the triangular lattice Hubbard model. Here we provide a simple picture for its origin in terms of a Bosonic RVB description. More specifically, we show that such a chiral spin liquid state can be understood as a quantum disordered tetrahedral spin state stabilized by the four spin ring exchange coupling, which suppresses the order-by-disorder effect toward a stripy spin state. Such a chiral spin liquid state features a spin Berry phase of $\fracπ{2}$ per triangle. However, we show that …
We investigate the $J_1$-$J_2$ Heisenberg model on the triangular lattice with an additional scalar chirality term and show that a chiral spin liquid is stabilized in a sizeable region of the phase diagram. This topological phase is situated in between a coplanar $120^\circ$ Néel ordered and a non-coplanar tetrahedrally ordered phase. Furthermore we discuss the nature of the spin-disordered intermediate phase in the $J_1$-$J_2$ model. We compare the groundstates from Exact Diagonalization with a Dirac spin liquid wavefunction and propose a scenario where this wavefunction describes the quantum…
Superconducting and charge-ordered phases from Dirac quantum spin liquids on the triangular lattice
Center for Computational Quantum Physics, Flatiron Institute（Flatiron研究所计算量子物理中心）；Department of Physics, Harvard University（哈佛大学物理系）；Max Planck Institute for Solid State Research（马克斯·普朗克固体研究所）
We model the newly predicted high-$T_c$ superconducting candidates constructed by corner-shared trigonal bipyramidal complexes with an effective three-orbital tight-banding Hamiltonian and investigate the pairing symmetry of their superconducting states driven by electron-electron interactions. Our combined weak and strong coupling based calculations consistently identify the chiral $d+id$ superconductivity as the leading pairing symmetry in a wide doping range with realistic interaction parameters. This pairing state has nontrivial topological Chern-number and can host gapless chiral edge mod…
While a quantum spin liquid (QSL) phase has been identified in the $J_1$-$J_2$ Heisenberg model on a triangular lattice via numerical calculations, debate persists about whether or not such a QSL is gapped or gapless, with contradictory conclusions from different techniques. Moreover, information about excitations and dynamics is crucial for the experimental detection of such a phase. In this work, we use exact diagonalization to characterize signatures of a QSL phase on the triangular lattice through the dynamical spin structure factor $\mathcal{S}(q,ω)$ and Raman susceptibility $\mathcalχ(ω)…
By using Gutzwiller projected fermionic wave functions and variational Monte Carlo technique, we study the spin-$1/2$ Heisenberg model with the first-neighbor ($J_1$), second-neighbor ($J_2$), and additional scalar chiral interaction $J_χ{\bf S}_i \cdot ({\bf S}_j \times {\bf S}_k)$ on the triangular lattice. In the non-magnetic phase of the $J_1-J_2$ triangular model with $0.08 \lesssim J_2/J_1 \lesssim 0.16$, recent density-matrix renormalization group (DMRG) studies [Zhu and White, Phys. Rev. B {\bf 92}, 041105 (2015); Hu, Gong, Zhu, and Sheng, Phys. Rev. B {\bf 92}, 140403 (2015)] find a p…
The interplay between spin frustration and charge fluctuation gives rise to an exotic quantum state in the intermediate-interaction regime of the half-filled triangular-lattice Hubbard model, while the nature of the state is under debate. Using the density matrix renormalization group with SU(2)$_{\rm{spin}} \otimes $U(1)$_{\rm{charge}}$ symmetries implemented, we study the triangular-lattice Hubbard model defined on the long cylinder geometry up to circumference $W=6$. A gapped quantum spin liquid, with on-site interaction $9 \lesssim U / t \lesssim 10.75$, is identified between the metallic …
We study the effects of quantum fluctuations on a non-coplanar tetrahedral spin structure, which has a scalar chiral order, in the spin-1/2 multiple-spin exchange model with up to the six-spin exchange interactions on a triangular lattice. We find that, in the linear spin-wave approximation, the tetrahedral structure survives the quantum fluctuations because spin waves do not soften in the whole parameter region of the tetrahedral-structure phase evaluated for the classical system. In the quantum corrections to the ground-state energy, sublattice magnetization, and scalar chirality, the effect…
The existence of a gapped chiral spin liquid has been recently suggested in the vicinity of the metal-insulator transition of the Hubbard model on the triangular lattice, by intensive density-matrix renormalization group (DMRG) simulations [A. Szasz, J. Motruk, M.P. Zaletel, and J.E. Moore, Phys. Rev. X {\bf 10}, 021042 (2020)]. Here, we report the results obtained within the variational Monte Carlo technique based upon Jastrow-Slater wave functions, implemented with backflow correlations. As in DMRG calculations, we consider $N$-leg cylinders. For $N=4$ and in the presence of a next-nearest n…
Spin chirality ordering and anomalous Hall effect in the ferromagnetic Kondo lattice model on the triangular lattice
通过变分计算，我们研究了三角晶格上铁磁近藤晶格模型的基态相图。我们发现，除了3/4填充的情况外，在1/4填充处及其附近还出现了具有有限标量自旋手征的非共面四子晶格有序，这是由费米面的完美嵌套引起的。Martin和C. D. Batista：Phys. Rev. Lett. { \bf 101}（2008）156402]。1/4填充相在比3/4填充相更宽的参数范围内是稳定的，并且包括由陈数表征的大的带隙绝缘状态区域。我们还计算了手征有序相的霍尔电导率。
We study the interplay of competing interactions in spin-$1/2$ triangular Heisenberg model through tuning the first- ($J_1$), second- ($J_2$), and third-neighbor ($J_3$) couplings. Based on large-scale density matrix renormalization group calculation, we identify a quantum phase diagram of the system and discover a new {\it gapless} chiral spin liquid (CSL) phase in the intermediate $J_2$ and $J_3$ regime. This CSL state spontaneously breaks time-reversal symmetry with finite scalar chiral order, and it has gapless excitations implied by a vanishing spin triplet gap and a finite central charge…
At strong repulsion, the triangular-lattice Hubbard model is described by $s=1/2$ spins with nearest-neighbor antiferromagnetic Heisenberg interactions and exhibits conventional 120$^\circ$ order. Using the infinite density matrix renormalization group and exact diagonalization, we study the effect of the additional four-spin interactions naturally generated from the underlying Mott-insulator physics of electrons as the repulsion decreases. Although these interactions have historically been connected with a gapless ground state with emergent spinon Fermi surface, we find that at physically rel…
Thermodynamic properties of the twisted three-leg spin tube under magnetic field are studied by the finite-$T$ density-matrix renormalization group method. The specific heat, spin, and chiral susceptibilities of the infinite system are calculated for both the original and its low-energy effective models. The obtained results show that the presence of the chirality is observed as a clear peak in the specific heat at low temperature and the contribution of the chirality dominates the low-temperature part of the specific heat as the exchange coupling along the spin tube decreases. The peak struct…
Motivated by experimental studies that have found signatures of a quantum spin liquid phase in organic crystals whose structure is well described by the two-dimensional triangular lattice, we study the Hubbard model on this lattice at half filling using the infinite-system density matrix renormalization group (iDMRG) method. On infinite cylinders with finite circumference, we identify an intermediate phase between observed metallic behavior at low interaction strength and Mott insulating spin-ordered behavior at strong interactions. Chiral ordering from spontaneous breaking of time-reversal sy…
We study ground-state properties of a triangular triple quantum dot connected to two superconducting (SC) leads. In this system orbital motion along the triangular configuration causes various types of quantum phases, such as a Kondo effect with a four-fold degenerate state and the Nagaoka ferromagnetic mechanism, depending on the electron filling. The ground state also evolves as the Cooper pairs penetrate from the SC leads. We describe the phase diagram in a wide range of the parameter space, varying the gate voltage, the couplings between the dots and leads, and also the Josephson phase bet…
The mechanism of the unconventional topological superconductivity (TSC) remains a long-standing issue. We investigate the quantum phase diagram of the extended $t$-$J$-$J_χ$ model including spin chiral interactions on triangular lattice based on the state-of-the-art density matrix renormalization group simulations. We identify distinct classes of superconducting phases characterized by nonzero topological Chern numbers $C=1$ and $2$, and a nematic d-wave superconducting phase with a zero Chern number. The TSC states are shown to emerge from doping either a magnetic insulator or chiral spin liq…
Chiral Magnetism and Quantum Anomalous Hall Effect in a Low-energy Kondo Model on the Triangular Lattice
University of California Santa Barbara（加州大学圣塔芭芭拉分校）；Kavli Institute for Theoretical Physics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校卡弗里理论物理研究所）；French American Center for Theoretical Science, CNRS, KITP, Santa Barbara, California（法国美国理论科学中心）；Canadian Institute for Advanced Research（加拿大高级研究院）
Chiral enhancement of two-magnon bound states in an $S=1/2$ triangular-lattice magnet
Budapest University of Technology and Economics（布达佩斯技术与经济大学）；HUN-REN Wigner Research Centre for Physics, Institute for Solid State Physics and Optics（匈牙利研究网络维格纳物理研究中心固体物理与光学研究所）
Single helicity of the triple-$q$ triangular skyrmion lattice state in cubic chiral helimagnet EuPtSi
Hiroshima University（广岛大学）；Japan Atomic Energy Agency（日本原子能机构）；Institute of Materials Structure Science, High Energy Accelerator Research Organization（高能加速器研究机构物质结构研究所）；University of the Ryukyus（琉球大学）；RIKEN Center for Emergent Matter Science（理化学研究所 新兴物质科学中心）
Skyrmions, topologically-protected nanometric spin vortices, are being investigated extensively in various magnets. Among them, many of structurally-chiral cubic magnets host the triangular-lattice skyrmion crystal (SkX) as the thermodynamic equilibrium state. However, this state exists only in a narrow temperature and magnetic-field region just below the magnetic transition temperature $T_\mathrm{c}$, while a helical or conical magnetic state prevails at lower temperatures. Here we describe that for a room-temperature skyrmion material, $β$-Mn-type Co$_8$Zn$_8$Mn$_4$, a field-cooling via the …
We study topography of the spin-polarized bound states of magnetic impurities embedded in triangular lattice of a superconducting host. Such states have been observed experimentally in 2H-NbSe$_{2}$ crystal [G. C. M'enard et al., Nat.Phys. 11, 1013 (2015)] and revealed the oscillating particle-hole asymmetry extending to tens of nanometers. Using the Bogoliubov-de Gennes approach we explore the Yu-Shiba-Rusinov states in presence of spin-orbit interaction. We also study the bound states of double impurities for several relative positions in a triangular lattice.
We study the ground state properties of the Hubbard model on three-leg triangular cylinders using large-scale density-matrix renormalization group simulations. At half-filling, we identify an intermediate gapless spin liquid phase between a metallic phase at weak coupling and Mott insulating dimer phase at strong interaction, which has one gapless spin mode and algebraic spin-spin correlations but exponential decay scalar chiral-chiral correlations. Upon light doping the gapless spin liquid, the system exhibits power-law charge-density-wave (CDW) correlations but short-range single-particle, s…
Chiral Spin Liquid and Quantum Phase Transition in the Triangular Lattice Hofstadter-Hubbard Model
University of California, Berkeley（加州大学伯克利分校）；Center for Computational Quantum Physics, Flatiron Institute（Flatiron Institute计算量子物理中心）；Harvard University（哈佛大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Columbia University（哥伦比亚大学）
SU($N$) Mott insulators have been proposed and/or realized in solid-state materials and with ultracold atoms on optical lattices. We study the two-dimensional SU($N$) antiferromagnets on the triangular lattice. Starting from an SU($N$) Heisenberg model with the fundamental representation on each site in the large-$N$ limit, we perform a self-consistent calculation and find a variety of ground states including the valence cluster states, stripe ordered states with a doubled unit-cell and topological chiral spin liquids. The system favors a cluster or ordered ground state when the number of flav…
