Gapless Kitaev Spin Liquid to Classical String Gas through Tensor Networks

We provide a framework for understanding the gapless Kitaev spin liquid (KSL) in the language of the tensor network (TN). Without introducing a Majorana fermion, most of the features of the KSL, including the symmetries, gauge structure, criticality, and vortex freeness, are explained in a compact TN representation. Our construction reveals a hidden string gas structure of the KSL. With only two variational parameters to adjust, we obtain an accurate KSL Ansatz with the bond dimension $D=8$ in a compact form, where the energy is about 0.007% higher than the exact one.

Figure 1

Schematic figures of (a) the TPS setup on the honeycomb lattice, (b) its building block tensor $T_{ijk}^s$, and (c) a building block tensor $Q_{ijk}^{s{s}^{\prime }}$ of the loop gas TPO defined in Eq. (2). Here, the $x$, $y$, and $z$ links in the model [Eq. (1)] are characterized in red, blue, and yellow, respectively.

Figure 2

Overview of the $n$th order Ansatz $|{\psi }_n⟩$ obtained by LG and DG operators, where SG denotes the string gas, $dE=E-E_{\text{Kitaev}}$ denotes the energy deviation, and $c$ stands for the central charge.

Figure 3

(a) Energy of $|{\psi }_1(\varphi )⟩$, whose building block tensor is defined in Eq. (9), and (b) the central charge $c$ and scaling dimensions (${\mathrm{\Delta }}_i$) as a function of $\varphi$. The black solid lines in (b) denote the exact $c$ and ${\mathrm{\Delta }}_i$ from the Ising universality class.

Figure 4

(a) Energy landscape of $|{\psi }_2(\varphi ,\theta )⟩$ constructed by the tensor in Eq. (10) as functions of $\varphi$ and $\theta$. The energy minimum is denoted by the black dot, at which the variational energy is $E=-\mathrm{0.196\; 81}$ [33]. (b) The five largest correlation lengths ${\xi }_i$ of the best Ansatz represented by the dot in (a), where $\chi$ stands for the bond dimension of CTMRG.