Theory of network contractor dynamics for exploring thermodynamic properties of two-dimensional quantum lattice models

Based on the tensor network state representation, we develop a nonlinear dynamic theory, coined network contractor dynamics (NCD), to explore the thermodynamic properties of two-dimensional quantum lattice models. By invoking the rank-1 decomposition in the multilinear algebra, the NCD scheme makes the contraction of the tensor network of the partition function be realized through a contraction of a local tensor cluster with vectors on its boundary. An imaginary-time-sweep algorithm for implementation of the NCD method is proposed for practical numerical simulations. We benchmark the NCD scheme on the square Ising model, which shows great accuracy. Also, the results on the spin-1/2 Heisenberg antiferromagnet on a honeycomb lattice are disclosed to be in good agreement with the quantum Monte Carlo calculations. The quasientanglement entropy $S$, Lyapunov exponent $I^{\text{Lya}}$, and loop character $I^{\text{loop}}$ are introduced within the dynamic scheme, which are found to display “nonlocality” near the critical point, and can be applied to determine the thermodynamic phase transitions of both classical and quantum systems.

Figure 1

(a) By contracting three ${\mathbf{G}}^L$'s and ${\mathbf{G}}^L$'s together, respectively, we obtain the two inequivalent tensors $\mathbf{A}$ and $\mathbf{B}$ of the TPDO as shown in (b). (c) The TPDO is evolved by contracting a pair of ${\mathbf{G}}^L$ and ${\mathbf{G}}^R$ to the tensors $\mathbf{A}$ and $\mathbf{B}$. (d) After tracing over the physical bonds, we obtain the cell tensor ${\mathbf{T}}^{\text{cell}}$ by contracting a shared bond of $\mathbf{A}$ and $\mathbf{B}$. (e) The bond $g_3$ of the left ${\mathbf{T}}^{\text{cell}}$ connects the bond $g_1$ of the right ${\mathbf{T}}^{\text{cell}}$, thus $g_1$ and $g_3$ form an index pair.

Figure 2

(a) The infinite defective TN of the partition function $Z$ can be contracted as a tree TN, because the $\tilde{\mathbf{T}}$'s (the rank-1 approximation of ${\mathbf{T}}^{\text{cell}}$ formed by the direct product of contractors) break the loops of the original TN, as shown by the shaded area. By repeatedly using the fixed point conditions [Eq. (5)], this contraction leads to Eq. (7), giving a local contraction of ${\mathbf{T}}^{\text{cell}}$ and contractors. (b) Another possible construction of the defective TN with a tensor cluster that contains loops (yellow areas) of ${\mathbf{T}}^{\text{cell}\left(1\right)}$'s.

Figure 3

(a) The contractors satisfy the fixed point condition [Eq. (5)]. Due to the permutation invariance, when $i$ and $j$ belong to an index pair, the contractors satisfy ${\mathbf{x}}^{{\alpha }_i}={\mathbf{x}}^{{\alpha }_j}$. (b) The thermal average of an operator ${\widehat{O}}_i$ [Eq. (8)]. (c) The environment matrix of the enlarged bond $\tilde{g}$ [Eq. (15)].

Figure 4

(a) A sketch of increasing the size of the cell tensor. Three ${\mathbf{T}}^{\text{cell}\left(1\right)}$'s are contracted to get ${\mathbf{T}}^{\uparrow }$ and ${\mathbf{T}}^{\downarrow }$ (dashed circles), and then by Eq. (10), the whole cluster is considered as a cell tensor with $\gamma =12$. The green blocks with blue bonds represent corresponding contractors. (b) From the SVD of the contractor ${\mathbf{x}}^{{\alpha }_{a\left(c\right)},\gamma }$, we have the singular value spectrum $\mu$, and the QEE can be obtained with $\mu$ through Eq. (11).

Figure 5

The flowchart of the ITSA algorithm.

Figure 6

The temperature dependence of the relative error $\Delta f=|(f_{\text{NCD}}-f_{\text{exact}})/f_{\text{exact}}|$ of the free energy of the 2D Ising model on square lattice calculated by the NCD ($\chi =16,24,32$), TRG, SRG, and HOSRG algorithms (Ref. 25) ($\chi =24$). We set the cell tensor size $\gamma \sim {10}^5$ for the NCD calculations.

Figure 7

(a) The temperature dependence of the QEE $S$ of the 2D Ising model on a square lattice, where the critical temperature $T^S=2.271$ is obtained. Inset: The temperature dependence of the loop character, showing $T^{\text{loop}}=2.270$. (b) The temperature dependence of the Lyapunov exponent of the 2D Ising model, giving $T^{\text{Lya}}=2.271$. The inset shows the detail near the critical point. The relative error of the critical temperature is about ${10}^{-3}$, where the exact critical temperature $T_c=2/\ln (1+\sqrt{2})\simeq 2.2692$.

Figure 8

(a) Energy per site $E$ obtained by NCD and QMC of the spin-$1/2$ Heisenberg antiferromagnet on a honeycomb lattice at $\Delta =0.5$ and 1. Inset: The energy difference $\Delta E=|E_{\text{NCD}}-E_{\text{QMC}}|$ against temperature at $\chi =12$ and $\chi =16$. (b) The staggered magnetization per site $m_z$ and the specific heat $C$ as functions of temperature $T$ for $\Delta =0.5$. The specific heat indicates a thermodynamic phase transition at $T^C=0.345$.

Figure 9

(a) The temperature dependence of the QEE $S$, where the critical temperature is found to be $T^S=0.3610$. (b) The temperature dependence of the Lyapunov exponent $I^{\text{Lya}}$ and the loop character $I^{\text{loop}}$, which show sharp peaks at $T^{\text{Lya}}=0.3636$ and $T^{\text{loop}}=0.3597$, respectively.