Single-layer tensor network study of the Heisenberg model with chiral interactions on a kagome lattice

We study the antiferromagnetic kagome Heisenberg model with an additional scalar-chiral interaction by using the infinite projected entangled-pair state (iPEPS) ansatz. We discuss in detail the implementation of the optimization algorithm in the framework of the single-layer tensor network based on the corner transfer matrix technique. Our benchmark based on the full-update algorithm shows that the single-layer algorithm is stable, which leads to the same level of accuracy as the double-layer ansatz but with much less computation time. We further apply this algorithm to study the nature of the kagome Heisenberg model with a scalar-chiral interaction by computing the bond dimension scaling of magnetization, bond energy difference, chiral order parameter, and correlation length. In particular, we find that for strong chiral coupling the correlation length, which is extracted from the transfer matrix, saturates to a finite value for large bond dimension, representing a gapped spin-liquid state. Further comparison with density matrix renormalization group results supports that our iPEPS faithfully represents the time-reversal symmetry breaking chiral state. Our iPEPS simulation results shed new light on constructing PEPS for describing gapped chiral topological states.

Figure 1

(a) Tensor-network representation of the iPEPS $|\mathrm{\Psi }\rangle$, made from local five-rank tensors $\left\{A\right\}$ (solid circles). The virtual bonds have bond dimension $D$. (b) The tensor $a$ is defined by contraction on the physical index of tensors $A$ and $A^{†}$. In tensor $a$, virtual bonds have bond dimension $D^2$. (c) The scalar product $\langle \mathrm{\Psi }|\mathrm{\Psi }\rangle$ is obtained by contracting an infinite double-layer tensor network $\mathcal{E}$. (d) A finite tensor network $\mathcal{G}$, made from environment tensors $E_1,\cdots ,E_8$ that are obtained by the CTM approach, to approximate tensor network $\mathcal{E}$. The boundary bond dimension $\chi$ controls accuracy of this approximation. (e) The one-layer tensor network $\mathcal{M}$ represents the scalar product $\langle \mathrm{\Psi }|\mathrm{\Psi }\rangle$, obtained without tracing over the physical index as before. The right figure, representing a $2\times 2$ unit cell tensor network, is driven from the left one by reshaping physical bonds and inserting identity tensors $e$ and $e^{\prime }$ for crossing bonds. (f) The tensor network $\mathcal{M}$, made from tensors $\{A,A^{†},e,e^{\prime }\}$, could be approximated by using a standard CTM approach adapted to a $2\times 2$ unit cell.

Figure 2

(a) The one-layer tensor network representation of the scalar product $\langle \psi |\psi \rangle$, where $|\psi \rangle$ represent a $2\times 2$ unit cell iPEPS, made of tensors $\{a,b,c,d\}$. (b) To approximate that, we need to use a standard CTM approach adapted to a $4\times 4$ unit cell, made from tensors $\{a,b,c,d,a^{†},b^{†},c^{†},d^{†},e,e^{\prime }\}$. (c) Tensor-network diagram of reduced-tensor application: Tensors $\{a,b\}$ are decomposed to low-rank tensors $\{l,r,Q,Q^{\prime }\}$ by using LQ and QR decomposition. (d) Tensor-network representation of norm tensor $\mathcal{N}$. Right panel is obtained from the left one by using reduced-tensor application and removing tensors $\{l,r,l^{†},r^{†}\}$. The norm tensor $\mathcal{N}$ obtained by contracting the whole right network, so that the optimal computation time scales like $O\left(D^9\right)$, achieved by a left-to-right step-by-step contraction.

Figure 3

A comparison between single- and two-layer tensor-network ansatz. (a) The variational energy $E$ as a function of computational time (seconds) in log scale for imaginary time $\tau =0.2$ and bond dimension $D=5$. (b) The iteration number in the CTM approach $N_{\text{CTM}}$ (to meet a threshold value) versus bond dimension $D$. (c) The characteristic correlation lengths $\xi$, obtained by transfer matrix, as a function of boundary bond dimension $\chi$ for both single-layer and two-layer tensor-network ansatz. (d) The linear-log plot of spin-spin (only $z$ component) correlation function versus distant $r$.

Figure 4

Tensor-network representation of transfer matrix. (a) ${\mathcal{T}}_d$ and (b) ${\mathcal{T}}_d$ provide two different representations to estimate correlation length of the system, with different computational cost and accuracy. In the $\chi \to \infty$ limit, estimated correlation lengths should converge to the same value ${\xi }_d\to {\xi }_s$. ${\mathcal{T}}_d$ and ${\mathcal{T}}_s$ are respectively used in the double-layer and single-layer tensor-network ansatz.

Figure 5

(a) The iPEPS variational ground-state energy $E$, obtained by using full and simple update in a single-layer tensor-network framework, versus bond dimension $1/D$ at point $J_{ch}=0.5$. DMRG result is for a cylinder with width $L_y=12$. (b) The magnetic order parameter $m$ as a function of bond dimension $1/D$ at points $J_{ch}=0,0.5$. The dashed lines stand for an algebraic fit $m\sim D^{-\beta }$. Our estimation of the exponent $\beta$ is 0.6 and 5 respectively for points $J_{ch}=0$ and $J_{ch}=0.5$ by using full update. (c) The lattice symmetry breaking parameter $\mathrm{\Delta }T$ and (d) the scalar chirality term $H_c$ as a function of bond dimension $1/D$.

Figure 6

(a), (b) The correlation length $\xi$, extracted from transfer matrix ${\mathcal{T}}_s$, as a function of boundary bond dimension $\chi$ for $J_{ch}=2,0.5$. (c) The correlation length $\xi$ versus bond dimension $D$ for $J_{ch}=0.5,2,\infty$. Here $\xi$ has been extrapolated to the infinite-$\chi$ limit. (d) The linear-log plot of spin-spin and dimer-dimer correlation functions versus distance $r$ for $J_{ch}=0.5$. The slopes show inverse of the associated spin and dimer correlation length ${\xi }_s^{-1}$ and ${\xi }_d^{-1}$.

Figure 7

The scalar chiral order $\langle H_c\rangle$ and its derivative as a function of coupling parameter $J_{ch}$ for different bond dimensions. The DMRG results are obtained on the $L_y=8$ cylinder system.