Recent progress in quantum spin liquids, fractional magnetization plateaus, and unconventional superconductivity in kagome lattices

TL;DR

This article surveys recent progress on quantum spin liquids and unconventional superconductivity in kagome lattices. It integrates theoretical frameworks such as gauge theory and variational Monte Carlo with experimental observations in kagome magnets and AV3Sb5 superconductors. Key findings include evidence for a Dirac spin liquid ground state, field-induced magnetization plateaus consistent with quantum spin states, and signatures of chiral superconductivity and pairing density waves in AV3Sb5. The review highlights how geometry, electronic structure, and strong correlations conspire to produce novel quantum phases and outlines directions for resolving open questions.

Abstract

The kagome lattice, with its unique geometric structure, has emerged as a leading platform for exploring quantum many-body physics, particularly in the study of quantum spin liquids (QSLs) and unconventional superconductivity. This review highlights recent advancements in the investigations of QSLs, fractional magnetization plateau phases in kagome antiferromagnets, and unconventional superconductivity in vanadium-based kagome superconductors. We begin by examining the classical ground-state properties of the nearest-neighbor kagome antiferromagnetic Heisenberg model and introducing recent experimental progress in the study of QSLs and fractional magnetization plateau phases. Next, we discuss the fermionic description of the QSL states, along with related gauge theory and the variational Monte Carlo (VMC) method. We then focus on discussing the VMC studies of QSLs and magnetization plateau phases in kagome antiferromagnets. For superconductivity in kagome systems, we first analyze the characteristics of the electronic structure and the possible associated electronic instabilities. Finally, we review recent experimental advances in unconventional superconductivity in AV$_3$Sb$_5$ (A = K, Rb, Cs), with a particular focus on chiral superconductivity and pairing density waves.

Figures (6)

• Figure 1: (a) Kagome lattice structure. Each unit cell consists of three sublattices, labeled A, B, and C. For the NN AF Heisenberg model, the system exhibits significant geometric frustration. As illustrated, when two adjacent spins in a unit cell are arranged antiparallel, the third spin encounters a dilemma. (b) Two representative spin configurations with the lowest energy for classical spins. Left: $\boldsymbol{Q} = 0$ state. Right: $\sqrt{3}\times\sqrt{3}$ state. (c) Structure of YCu$_3$(OH)$_6$Br$_2$[Br$_{1-x}$(OH)$_x$]. Cu$^{2+}$ ions form a kagome lattice, and Y$^{3+}$ ions locate at the centers of the stars. Br$^{-}$ are positioned above and below the Y$^{3+}$, randomly mixed with OH$^{-}$. O$^{2-}$ around the hexagons are alternately displaced above and below the kagome plane. Adapted from Ref. pnas.2421390122. (d) Inelastic neutron scattering spectra of YCu$_3$(OH)$_6$Br$_2$[Br$_{1-x}$(OH)$_x$] along the $[H, 0]$ direction at 0.3 K. Adapted from Ref. nat.phys.20.1097. (e) Schematic illustration showing two conical spinon Dirac cones (red) merging into a solid cone spin excitation with a continuum inside (blue). Adapted from Ref. nat.phys.20.1097. (f) Six conical low-energy spin excitations in YCu$_3$(OH)$_6$Br$_2$[Br$_{1-x}$(OH)$_x$], with their momenta indicated in the kagome Brillouin zone. Adapted from Ref. nat.phys.20.1097. (g) Magnetization of YCu$_3$(OH)$_6$Br$_2$[Br$_{1-x}$(OH)$_x$]. Top: Magnetization curve. Bottom: Differential magnetic susceptibility $dM/dH$. Adapted from Ref. pnas.2421390122.
• Figure 2: (a) Energy per site of the $\mathbb{Z}_{2}[0, \pi]\beta$ state as a function of the second NN pairing $\Delta_{2}$ when $J_{2}/J_{1} = 0.15$, with system size $L=4, 8$ and $12$ in left, middle and right panels, respectively. Adapted from Ref. PhysRevB.91.020402. (b) Spin dynamical structure factor $S^{z}(\boldsymbol{q}, \omega)$ of the Dirac QSL, calculated by VMC method. Adapted from Ref. SciPostPhys.14.139. (c) Left: Phase diagram of the $s = 1$$J_{1}$-$J_{2}$-$J_{\chi}$-$J_{r}$ AF Heisenberg model on kagome lattice, including FM order, $Z_{2}$-SL ($Z_{2}$ spin liquid), NOQF (nematic order with quantum fluctuations) and NACSL (non-Abelian chiral QSL). Here, $J_{2}$, $J_{\chi}$ and $J_{r}$ are biquadratic, three-spin chiral and ring exchange interactions, respectively. Right: Three-fold degenerate ground state of NACSL from the ED method. Adapted from Ref. PhysRevA.108.033308.
• Figure 3: (a) Magnetization curve of the spin-1/2 kagome Heisenberg antiferromagnet in a uniform magnetic field, calculated using the DMRG approach. Five notable magnetization plateaus are observed at values of $M/M_{s}$ ($M_{s} = 1/2$ is the saturated value for the spin-1/2 system): 0, 1/9, 1/3, 5/9, and 7/9, in addition to the full polarization at $M/M_{s}=1$. Adapted from Ref. nat.com.4.2287. (b) A VBS state proposed by tensor network numerical calculations at the 1/9 plateau. Adapted from Ref. PhysRevB.107.L220401. (c) A $\sqrt{3} \times \sqrt{3}$ VBS state proposed by ED simulations in a small cluster with 36 sites at the 1/9 plateau. Adapted from Ref. JPSJ.93.123706. (d) The ansätz with $2\pi/3$ flux threading each primitive cell of kagome lattice and uniform magnetization $M/M_{s} = 1/9$ of $\mathbb{Z}_{3}$ QSL as the ground state of 1/9 magnetization plateau, proposed by VMC study. Adapted from Ref. PhysRevLett.133.096501. (e) The magnetization distribution, labeled by VBS-$\alpha$, from DMRG calculations nat.com.4.2287 and the mean-field ansätze of the three degenerate $\sqrt{3} \times \sqrt{3}$ VBS states proposed by VMC calculations cpl.42.090704. Red and blue bonds represent two types of hopping terms with different magnitudes, and the dashed bonds represent hopping terms with a minus sign. Adapted from the Ref. cpl.42.090704.
• Figure 4: (a) Band structures for the NN tight-binding model on the kagome lattice. Adapted from Ref. PhysRevB.85.144402. (b) Left: Fermi surface at the upper van Hove filling. The colors represent the sublattice weights. Adapted from Ref. PhysRevB.85.144402. Right: Nesting properties of Fermi surface. The solid black lines with arrows represent the nesting property for particle-hole scattering, while the orange dashed lines with arrows indicate the nesting property for particle-particle scattering. For clarity, the nesting wave vector in the particle-particle channel is adjusted from $\bm{Q}$ to $\bm{Q}+\bm{G}$, where $\bm{G}$ is a reciprocal lattice vector. (c) Left: Crystal structure of AV$_3$Sb$_5$ (A=K,Rb,Cs). Right: Projection of the crystal structure along the $c$-direction, showing V atoms forming a perfect kagome lattice. Adapted from Ref. Nat.Phys.18.137. (d) ARPES-measured Fermi surfaces (left) compared with DFT calculations (right) for CsV$_3$Sb$_5$. Adapted from Ref. PhysRevLett.125.247002. (e) Left: ARPES-measured band structures. Adapted from Ref. PhysRevLett.125.247002. Right: Band structures from DFT calculations for CsV$_3$Sb$_5$. Adapted from Ref. PhysRevB.105.235145. (f) Band structures of a minimal two-orbital tight-binding model. Adapted from Ref. PhysRevLett.127.177001.
• Figure 5: (a) Pressure-temperature phase diagram of CsV$_3$Sb$_5$ determined by $^{51}$V NMR measurements. Adapted from Ref. nature.611.682. (b) Temperature dependence of the Knight shift $\Delta K$ of $^{121}$Sb for CsV$_3$Sb$_5$ with $H\| a$. Adapted from Ref. cpl.38.077402. (c) Temperature dependence of $^{121}$($1/T_1T$) (left axis) and $^{123}$($1/T_1T$) (right axis) for CsV$_3$Sb$_5$. A Hebel–Slichter coherence peak appears just below $T_{\mathrm{c}}$. Adapted from Ref. cpl.38.077402. (d) Two kinds of SC spectra measured on half-Cs surface for CsV$_3$Sb$_5$. SC spectrum of Al polycrystal ($T_{\mathrm{c}}\sim1.2$ K) is displayed for comparison. Adapted from Ref. PhysRevLett.127.187004. (e), (f) SC energy gap structures measured by Bogoliubov quasiparticle interference for CsV$_3$Sb$_5$ and CsV$_{3-x}$Ti$_x$Sb$_5$ ($x\sim0.18$). Adapted from Ref. SCPMA.67.277411. (g) SC energy gap structures measured by ARPES for Cs(V$_{0.86}$Ta$_{0.14}$)$_3$Sb$_5$. Top: Positions of the Fermi wave vectors. Bottom: Gap amplitudes. Adapted from Ref. nature.617.488.