Multiscale entanglement renormalization ansatz for spin chains with continuously varying criticality

We use the multiscale entanglement renormalization ansatz (MERA) to numerically investigate three critical quantum spin chains with $Z_2\times Z_2$ on-site symmetry: a staggered XXZ model, a transverse field cluster model, and the quantum Ashkin-Teller model. All three models possess a continuous one-parameter family of critical points. Along this critical line, the thermodynamic limit of these models is expected to be described by classes of $c=1$ conformal field theories (CFTs) of two possible types: the $S^1$ free boson and its $Z_2$ orbifold. Our numerics using MERA with explicitly enforced $Z_2\times Z_2$ symmetry allow us to extract conformal data for each model, with strong evidence supporting the identification of the staggered XXZ model and critical transverse field cluster model with the $S^1$ boson CFT, and the Ashkin-Teller model with the $Z_2$-orbifold boson CFT. Our first two models describe the phase transitions between symmetry-protected topologically ordered phases and trivial phases, which lie outside the usual Landau-Ginsburg-Wilson paradigm of symmetry breaking. Our results show that a range of critical theories can arise at the boundary of a single symmetry protected phase.

Figure 1

Illustrative phase space diagrams and renormalization group (RG) flows for models with a unique RG fixed point (left) and a line of criticality (right). RG relevant operators have $\Delta <2$ and grow under renormalization, while those with $\Delta >2$ are called irrelevant, since a CFT deformed by such an operator flows back to the fixed point. Deformations by an exactly marginal operator lead to a new conformal fixed point. As the size of the deformation is varied, the conformal dimensions vary continuously.

Figure 2

Each of the spin models is defined on a pair of parallel chains. The ${\sigma }_{}^{}$ operators act only on the red (top) chain and the ${\tau }_{}^{}$ act only on the blue (bottom). The green region indicates a single site. Each of the models is then defined by a 1D nearest-neighbor Hamiltonian.

Figure 3

Tensor network diagram for the scale-invariant ternary MERA. Isometries are indicated by triangles (green), while disentanglers are rectangles (blue). The layered structure accurately represents a wide range of length scales and captures the scale-invariant physics of critical spin chains.

Figure 4

The descending superoperator. This computes the reduced density matrix of the model at level $\ell$ of the RG. This is just one of the contractions which must be performed to optimize the MERA tensors. The cost is $\mathcal{O}\left({\chi }^8\right)$, however, this can be reduced as discussed in the text.

Figure 5

Ground-state energies (GSE) extracted from the MERA and their relative errors $\Delta E$. For the bond dimension used here, these remain $\mathcal{O}\left({10}^{-4}\right)$ as the Hamiltonian parameter $\lambda$ is varied. The numerical GSE remains an upper bound on the true GSE. Note that the CFT radius $R_C$ is related to the spin model coupling parameter via Eq. (4).

Figure 6

Central charges extracted from MERA simulations. The CFTs associated with these models are expected to be $c=1$ (shown as solid line) for the full range of $R_C^2$ considered here.

Figure 7

The lowest scaling dimensions (${\Delta }_{\varphi }$) of the $(1,-1)$ charge sector of the three models. The XXZ and transverse field cluster models agree well with the $S^1$ boson, while the AT model is consistent with the fixed scaling operator from the orbifold theory. Note that the $S^1$ and orbifold CFT radii are related via Eq. (11).

Figure 8

The behavior expected from the $S^1$ CFT is recovered by the XXZ and TFCM (a) and (b), while AT simulations recover the orbifold behavior (c) and (d). The accuracy of the recovered scaling dimensions ${\Delta }_{\varphi }$ degrades for larger ${\Delta }_{\varphi }$. Note that the $S^1$ and orbifold CFT radii are related via Eq. (11).

Figure 9

Results of an exact diagonalization study of the Ashkin-Teller model in the symmetric $(1,1)$ sector. (a) The energy gap for a range of values of $R_O$ and system sizes $L$. The gap closes as we approach the thermodynamic limit. The gap is plotted for various $1/R_O^2$ values; in blue ($\times$) is the point $1/R_O^2=1$ and in magenta ($+$) the point $1/R_O^2=2$. Lines represent a linear fit to the ED data. (b) A comparison of the finite-size scaling results for $L=12$ with the MERA and the CFT, for the $Z_2\times Z_2$ symmetric sector.

Figure 10

Scaling dimensions of nonlocal operators extracted from MERA simulations. These correspond to the spectrum of spin chains with boundary conditions twisted by the group elements defined in Eq. (9).

Figure 11

Only the $Z_2\times Z_2$ subgroup of the $D_4$ symmetry of the Ashkin-Teller model is enforced in the MERA. This can lead to coupling between scaling operators which one would expect to be forbidden, in turn leading to avoided crossings in the conformal dimensions. This data have ${\chi }_l=20=\overline{\chi }/4,{\chi }_u=16$, and four transitional layers.