Doubling of entanglement spectrum in tensor renormalization group

We investigate the entanglement spectrum in HOTRG—tensor renormalization group (RG) method combined with the higher order singular value decomposition—for two-dimensional (2D) classical vertex models. In the off-critical region, it is explained that the entanglement spectrum associated with the RG transformation is described by “doubling” of the spectrum of a corner transfer matrix. We then demonstrate that the doubling actually occurs for the square-lattice Ising model by HOTRG calculations up to $D=64$, where $D$ is the cutoff dimension of tensors. At the critical point, we also find that a nontrivial $D$ scaling behavior appears in the entanglement entropy. We mention about the HOTRG for the 1D quantum system as well.

Figure 1

Diagram for a local vertex weight, where $x,x^{\prime },y,y^{\prime }$ represent two-state link variables.

Figure 2

Schematic diagram of the vertex tensor $W_{x_{\mathrm{a}}{x}_{\mathrm{a}}^{\prime }{y}_{\mathrm{a}}{y}_{\mathrm{a}}^{\prime }}^{\left(n\right)}$ on a $L\times L$ lattice with $L=2^n$. The indices of the tensor specify the configurations at the edges.

Figure 3

Graphical representation of the reduced density matrix in Eq. (2).

Figure 4

Double line representation of the renormalized vertex tensor. A solid triangle at a corner indicates a CTM, and the link variable $\mu$ is represented by a combination of the two sub-link variables ${\mu }_1$ and ${\mu }_2$.

Figure 5

Double line representation of the reduced density matrix ${\rho }^{*}$ in Eq. (8). The closed loops indicate trace of the products of CTMs, which give scalar constants.

Figure 6

Graphical representation of the $L$-layer transfer matrix $\mathcal{T}$ in Eq. (14).

Figure 7

Entanglement spectrum ${\Omega }_{\mu }^{*}$ of the reduced density matrix ${\rho }^{*}$ for $D=49$. The open circles with the solid and broken lines, respectively, show the doubling spectra of the corresponding CTMs.

Figure 8

Absolute value of relative error $\varepsilon (L,D)$ in Eq. (19) at $T_{\mathrm{c}}$. The solid line shows the fitting to $\left|\varepsilon \right|\sim a{L}^b$, where the best fit is obtained for $a=0.69$ and $b=-2.00$.

Figure 9

Relative error $\varepsilon (L,D)$ in Eq. (19) at $T_{\mathrm{c}}$ in the large-$L$ limit. The solid line shows a fitting line $a{D}^{-4}$ with $a=0.034$.

Figure 10

Entanglement entropy in $S(L,D)$ at $T_{\mathrm{c}}$. The solid line shows the linear fit to the function $(c/3)logL+b$.

Figure 11

The $D$ dependence of $S(\infty ,D)$ at $T_{\mathrm{c}}$. The line represent the least square fitting to the function $(\theta /6)\log D+b$.

Figure 12

The entanglement spectrum ${\Omega }^{*}$ with $D=49$ at $T_{\mathrm{c}}$. For comparison, a nontrivial power of the doubling of the CTM spectrum ${\Lambda }^{*}$ obtained by a CTMRG calculation under the condition $m=7$ is presented as a solid line with circles.

Figure 13

System size dependence of $\mathrm{rank}\Omega$ in Eq. (3), where ${\Omega }_{\mu }/{\Omega }_1<{10}^{-15}$ is regarded as zero numerically.