Fermionic algebraic quantum spin liquid in an octa-kagome frustrated antiferromagnet

We investigate the ground state and finite-temperature properties of the spin-1/2 Heisenberg antiferromagnet on an infinite octa-kagome lattice by utilizing state-of-the-art tensor network-based numerical methods. It is shown that the ground state has a vanishing local magnetization and possesses a $1/2$-magnetization plateau with an up-down-up-up spin configuration. A quantum phase transition at the critical coupling ratio $J_d/J_t=0.6$ is found. When $0<J_d/J_t<0.6$, the system is in a valence bond state, where an obvious zero-magnetization plateau is observed, implying a gapful spin excitation; when $J_d/J_t>0.6$, the system exhibits a gapless excitation, in which the dimer-dimer correlation is found decaying in a power law, while the spin-spin and chiral-chiral correlation functions decay exponentially. At the isotropic point ($J_d/J_t=1$), we unveil that at low temperature $T$, the specific heat depends linearly on $T$, and the susceptibility tends to a constant for $T\to 0$, giving rise to a Wilson ratio around unity, implying that the system under interest is a fermionic algebraic quantum spin liquid.

Figure 1

Structure of the octa-kagome lattice (OKL). It can be obtained by stretching the triangles in a kagome lattice (KL) along the horizontal direction. If one stretches all three directions of the triangles in a KL, it will end up with a star lattice. The OKL can also be viewed as corner and edge sharing octagons. The blue balls represent spins sitting on the lattice site, and the red dashed parallelogram depicts a four-site unit cell, where ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are basis vectors. $J_t$ (black) and $J_d$ (orange) denote Heisenberg exchange couplings between nearest neighbor spins inside triangles and in-between triangles, respectively.

Figure 2

(a) The corresponding ground state TN representation on the OKL (black dashes). (b) The unit cell containing two nonequivalent tensors $A$ (red circle) and $B$ (green circle), and three different diagonal matrices ${\lambda }_1$ (blue diamond), ${\lambda }_2$ (pink diamond), and ${\lambda }_3$ (yellow diamond).

Figure 3

Graphical representation for the cluster update scheme. (a) The hexagonal cluster is composed of six tensors and twelve diagonal matrices. (b) We contract the physical indices and virtual bonds connected with diagonal matrices ${\lambda }_2$ and ${\lambda }_3$ of the double-layer tensor cluster to get tensor $W$, which will later be canonicalized for updating the diagonal matrices ${\lambda }_1$, as well as the tensors $A$ and $B$. (c) The orthogonal conditions for renewed ${\overline{\lambda }}_1$ and $\overline{W}$. (d) Extracting a physical index from the hexagonal cluster by Tucker decomposition.

Figure 4

(a) Singular value decomposition (SVD) of the double layer cluster in Fig. 3. (b) Permutation of physical indices from $A$ using Tucker decomposition. (c) Absorbing the physical indices into $B$.

Figure 5

(a) Unit cell of the TN in the NCD scheme, where ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are basis vectors. (b) The construction of $T^{\mathrm{cell}}$. (c) Transformation from $T^{\mathrm{cell}}$ to “defect.” The defect is constructed by six contractors $v^n$ ($n=1,2,...,6$) obtained by Eq. (6), and each contractor is denoted by a yellow circle connected with a bold black line.

Figure 6

The calculated energy per site for the ground state, $e_g$, versus inverse bond dimension $1/D_c$ (up to $D_c=10$ for NCD, $D_c=15$ for the cluster update, and $D_c=7$ for the full update). It is shown that $e_g$ decreases with enhancing $D_c$. Power law fittings for NCD (blue solid line), the cluster update (red dashed line), and the full update (black dash-dotted line) are given, and $e_g$ is extrapolated to be $-0.4517$ (NCD), $-0.4518$ (cluster update), and $-0.4524$ (full update) in the infinite $D_c$ limit.

Figure 7

The local magnetic moments (a) $\langle S_i^z\rangle$ and (b) $\langle S_i^x\rangle$ versus inverse bond dimension $1/D_c$ calculated by the cluster update, where $i=1,2,3,4$ denote four inequivalent sites in a unit cell marked in the inset of (a), and the “average” represents the intracellular mean magnetic moment. The local magnetic moments (c) $\langle S_i^z\rangle$ and (d) $\langle S_i^x\rangle$ versus inverse bond dimension $1/D_c$ calculated by NCD, where $i=1,2,3,4,5,6,7,8$ denote eight sites in an expanded cell in the inset of (c), and the average denotes the average of the magnetic moments over these eight sites.

Figure 8

Spatial dependence of correlation functions of the spin-1/2 HAFM on the OKL in (a) semilog and (b) log-log plots. The vertical direction is represented by subscript “$\perp$” and the horizontal direction is marked by “$=$”. In the vertical direction, the spin-spin (fitting with blue solid line), chiral-chiral (fitting with red dash-dotted line), and dimer-dimer (fitting with black dotted line) correlation functions show exponential decaying behaviors. In the horizontal direction, the spin-spin (fitting with blue short-dashed line) and chiral-chiral (fitting with red dash-dot-dotted line) correlation functions show exponential decaying behaviors, while the dimer-dimer (fitting with black dashed line) correlation function shows a power law decay. All correlation functions are calculated by the full update algorithm with $D_c=5$.

Figure 9

Magnetic curves of the spin-1/2 HAFM on the OKL. (a) $J_d=0.2,0.8$, and 1, (b) $J_d=0.1,0.4$, and 0.55 under low magnetic fields. Inset of (a): Up-down-up-up (UDUU) spin configuration in a unit cell in the $M=1/4$ plateau phase.

Figure 10

Spin gap as a function of $J_d$ for the spin-1/2 HAFM on the OKL. The inset is the second-order derivative of the ground state energy with respect to $J_d$ in the absence of a magnetic field. It is clear that the point $J_d=0.6$ is singular, at which the spin gap closes.

Figure 11

The ground state phase diagram for the spin-1/2 HAFM on the OKL. A quantum phase transition from the VBS phase to the QSL phase happens at the critical point $J_d^c=0.6$. UDUU: the 1/2-magnetization plateau phase with an up-down-up-up spin configuration.

Figure 12

Specific heat $C\left(T\right)$ versus temperature $T$ of the spin-$1/2$ HAFM on the OKL for $J_d={10}^{-4}$ (blue solid circles) and $J_d=1$ (red open circles). Inset: the low-temperature part of $C\left(T\right)$. For $J_d=1$ below $T=0.25,C\left(T\right)$ can be well fitted by a polynomial $C\left(T\right)=1.741T-7.96T^2+22.51T^3-25.61T^4$ (black line), and for $J_d={10}^{-4}$ below $T=0.8,C\left(T\right)$ is also better compared with an exact diagonalization (ED) result (cyan dashes) of the zig-zag spin chain containing eight triangles. Here the bond dimension is $D_c=20$.

Figure 13

Zero-field magnetic susceptibility $\chi$ as a function of temperature $T$ of the spin-$1/2$ HAFM on the OKL for $J_d=1$ (red open circles) and $J_d={10}^{-4}$ (blue solid circles). Inset: the low-temperature part of susceptibility, where the case of $J_d=1$ can be fitted with a polynomial $\chi \left(T\right)=0.09566-0.06828T+2.438T^2$ (black line), and that of $J_d={10}^{-4}$ behaves in an exponential way. Here the bond dimension is $D_c=20$.