Linearized Tensor Renormalization Group Algorithm for the Calculation of Thermodynamic Properties of Quantum Lattice Models

A linearized tensor renormalization group algorithm is developed to calculate the thermodynamic properties of low-dimensional quantum lattice models. This new approach employs the infinite time-evolving block decimation technique, and allows for treating directly the transfer-matrix tensor network that makes it more scalable. To illustrate the performance, the thermodynamic quantities of the quantum $XY$ spin chain as well as the Heisenberg antiferromagnet on a honeycomb lattice are calculated by the linearized tensor renormalization group method, showing the pronounced precision and high efficiency.

Figure 1

(a) A transfer-matrix tensor network, where each bond denotes the $\sigma$ index in Eqs. (2, 3). (b) A local transformation of a fourth-order tensor into two third-order tensors through a singular value decomposition (SVD). (c) Transform the transfer-matrix tensor network to a hexagonal one. (d) By contracting the intermediate bonds marked by dashed ovals in (c), one gets a brick wall structure with the fourth-order tensors in the bottom line.

Figure 2

A local evolution of the tensors by contraction and SVD. (a) Contract the intermediate bonds; (b) obtain a sixth-order tensor $O$; and (c) calculate the singular value decomposition (SVD) of $O$, and update the tensors $M_{a,b}$ and $\lambda$. The above manipulation has a computational cost that scales as $O(D^6{D}_c^3)$.

Figure 3

An successive projection of each row of tensors onto the MPO in the bottom line [(a)–(c)]. After the projection along the Trotter direction, by tracing out the physical indices $t$ and $b$ of the MPO, one may get a 1D matrix product, of which the trace can be obtained by a matrix RG procedure [(d)–(g)].

Figure 4

The relative error of the free energy per site, $\delta f$, of the quantum $XY$ spin chain at high temperatures. $\delta f$ converges rapidly with $D_c$, and the lines with $D_c=100$ and 150 coincide with each other ($\tau =0.05$, 0.02). In addition, the TMRG results ($\tau =0.1$, 0.05) are also presented for a comparison.

Figure 5

LTRG and TMRG results of the quantum $XY$ spin chain. (a) Relative error of the free energy per site $\delta f$. (b) The energy per site $e$. The inset shows the variation of $(e-e_0)/e_0$ with inverse temperature $\beta$ for various $D_c$.

Figure 6

(a) Specific heat as a function of temperature ($T=1/\beta$) of the quantum $XY$ spin chain. The inset shows the low temperature results for $D_c=100$ and 200, along with the TMRG data ($M=200$) for a comparison. (b) Energy per site of the 2D spin-$1/2$ Heisenberg antiferromagnet on a honeycomb lattice for different staggered magnetic fields. The QMC results are obtained by using the ALPS library [21].