Symmetry-respecting real-space renormalization for the quantum Ashkin-Teller model

We use a simple real-space renormalization-group approach to investigate the critical behavior of the quantum Ashkin-Teller model, a one-dimensional quantum spin chain possessing a line of criticality along which critical exponents vary continuously. This approach, which is based on exploiting the on-site symmetry of the model, has been shown to be surprisingly accurate for predicting some aspects of the critical behavior of the quantum transverse-field Ising model. Our investigation explores this approach in more generality, in a model in which the critical behavior has a richer structure but which reduces to the simpler Ising case at a special point. We demonstrate that the correlation length critical exponent as predicted from this real-space renormalization-group approach is in broad agreement with the corresponding results from conformal field theory along the line of criticality. Near the Ising special point, the error in the estimated critical exponent from this simple method is comparable to that of numerically intensive simulations based on much more sophisticated methods, although the accuracy decreases away from the decoupled Ising model point.

Figure 1

Real-space RG as a concatenated block code, presented as a tree tensor network. Here, the block size is $b=2$, and four layers of concatenation are shown.

Figure 2

Comparison of the critical index ${\nu }_{*}$ of the energy operator (black) with the critical exponent $\nu$ obtained from our real-space RG map (red) as functions of the Hamiltonian parameter $\lambda$, using the smallest blocking size $b=2$. Inset: relative error $|\nu -{\nu }_{*}|/{\nu }_{*}$ of the predicted critical exponent $\nu$ obtained from our real-space RG map to the critical index ${\nu }_{*}$.

Figure 3

Alternative choices of blocking. Within each block consisting of $b$ sites, one site (a pair of spins, such as shown encircled by dashed line), shown in red, is a preferred site for the encoding map.

Figure 4

Critical exponent $\nu$ obtained from our real-space RG map using various blocking sizes $b$, all with the preferred site $k_0=0$ on the edge of the block, as illustrated in Fig. 3. For comparison, the exact value ${\nu }_{*}$ is shown (black line). Inset: relative error $|\nu -{\nu }_{*}|/{\nu }_{*}$ of the predicted critical exponent $\nu$ obtained for each choice.

Figure 5

Critical exponent $\nu$ obtained from our real-space RG map using a blocking size $b=3$ with a symmetric (orange) vs asymmetric (blue) choice of internal Hamiltonian terms as illustrated in Fig. 3. For comparison, the exact value ${\nu }_{*}$ is shown (black line). Inset: relative error $|\nu -{\nu }_{*}|/{\nu }_{*}$ of the predicted critical exponent $\nu$ obtained for each choice.