Anomalies and entanglement renormalization

We study 't Hooft anomalies of discrete groups in the framework of (1+1)-dimensional multiscale entanglement renormalization ansatz states on the lattice. Using matrix product operators, general topological restrictions on conformal data are derived. An ansatz class allowing for optimization of MERA with an anomalous symmetry is introduced. We utilize this class to numerically study a family of Hamiltonians with a symmetric critical line. Conformal data is obtained for all irreducible projective representations of each anomalous symmetry twist, corresponding to definite topological sectors. It is numerically demonstrated that this line is a protected gapless phase. Finally, we implement a duality transformation between a pair of critical lines using our subclass of MERA.

Figure 1

The MERA represents a quantum state using layers of isometric tensors. Together, these tensors define a quantum circuit of logarithmic depth which can be used to prepare an entangled state from a product state. If the tensors are chosen appropriately, the network is thought to be able to accurately represent the ground state of gapless one-dimensional Hamiltonians. Throughout the paper we use a convention such that tensor network diagrams read bottom-to-top, corresponding to matrix multiplication read left-to-right.

Figure 2

By applying the MPO to a half-infinite chain, one can insert a domain wall between two dual theories. If the MPO acts as a symmetry, this corresponds to putting the theory on boundary conditions which have been twisted by the group element.

Figure 3

Phase diagram of the $abc$ model where $a+b+c=3$. SB = Symmetry breaking, ferromagnetic phase. SPT = ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry-protected topological phase. Para = Paramagnetic/disordered phase. RG fixed points are indicated in red, and the dashed blue lines indicate the unitary mappings between the phases. ISING = Ising duality map, KT = (Generalized) Kennedy-Tasaki transformation [98, 99], MPO = action of (1,1,1) defined in Eq. (36).

Figure 4

MERA data for the $abc$ model along the $b$ line. This line is symmetric under the full ${\mathbb{Z}}_2^3$. CFT data, including averaging process, is discussed in more detail in Appendix pp1. (a) The energy of the optimized MERA state. The state remains a ground state when the anomalous symmetry operator is applied. (b) Scaling dimensions of the associated CFT. These vary continuously with the parameter $b$. Points are averaged MERA data, while black lines correspond to Eq. (48a) for integers $e$ and $m$. Distinct colors/markers indicate under which irrep the fields transform. (c) Scaling dimensions of nonlocal operators corresponding to applying an anomalous symmetry [for group element $(1,1,1)$ defined in Eq. (36)] twist to half of the chain. Points are averaged MERA data, while black lines correspond to Eq. (48a) for $e,m\in \mathbb{Z}+1/2$. Distinct colors/markers indicate under which projective irrep the fields transform.

Figure 5

MERA data for the $abc$ model along the $a$ and $c$ lines. These are exchanged by the symmetry action. (a) Ground-state energy of the optimized MERA. By applying the symmetry operator to a state optimized for the Hamiltonian with $(a,b,b)$, we obtain a state which is the ground state of the Hamiltonian with parameters $(b,b,a)$. This demonstrates that the states are transforming properly. (b) The local fields in the CFTs describing these two lines as identical but distinct from those on the $b$ line.

Figure 6

Finite-size scaling data for the fully symmetric sector of the model. (a) After rescaling the spectrum so that the lowest excitation is consistent with the lowest nontrivial primary of the CFT, the fully symmetric states can be extracted. Fitting the data and extrapolating to the thermodynamic limit gives the scaling dimension. (b) For almost the whole $b$ line, we observe that there are no fully symmetric states with scaling dimension less than 2 (RG relevant). This implies that no local, symmetric, translationally invariant terms can be added to the Hamiltonian to gap it out; thus the gapless phase is protected.

Figure 7

MERA scaling dimensions for the trivial twist of the $abc$ model along the $b$ line. This line is symmetric under the anomalous action of ${\mathbb{Z}}_2^3$. Figure titles label: (twist label; irreducible representation label). Gray points are the raw data extracted from the MERA. Red points correspond to averaged data as discussed in Appendix pp1. Black lines correspond to local fields of the compactified free boson CFT.

Figure 8

Scaling dimensions for topological sectors with twists of the form $(x,y,0)$. Figure titles label: (twist label; irreducible projective representation label). Gray points are the raw data extracted from the MERA. Red points correspond to averaged data as discussed in Appendix pp1. Black lines correspond to equations in Table 1.

Figure 9

Scaling dimensions for topological sectors with twists of the form $(x,y,1)$. Figure titles label: (twist label; irreducible projective representation label). Gray points are the raw data extracted from the MERA. Red points correspond to averaged data as discussed in Appendix pp1. Black lines correspond to equations in Table 1.

Figure 10

The ternary MERA represents a quantum state using two types of tensors; unitary disentanglers (rectangles) and isometric tensors (triangles).

Figure 11

Identification of the edge degrees of freedom.