Exact projected entangled pair ground states with topological Euler invariant

Abstract

We report on a class of gapped projected entangled pair states (PEPS) with non-trivial Euler topology motivated by recent progress in band geometry. In the non-interacting limit, these systems have optimal conditions relating to saturation of quantum geometrical bounds, allowing for parent Hamiltonians whose lowest bands are completely flat and which have the PEPS as unique ground states. Protected by crystalline symmetries, these states evade restrictions on capturing tenfold-way topological features with gapped PEPS. These PEPS thus form the first tensor network representative of a non-interacting, gapped two-dimensional topological phase, similar to the Kitaev chain in one dimension. Using unitary circuits, we then formulate interacting variants of these PEPS and corresponding gapped parent Hamiltonians. We reveal characteristic entanglement features shared between the free-fermionic and interacting states with Euler topology. Our results hence provide a rich platform of PEPS models that have, unexpectedly, a finite topological invariant, forming the basis for new spin liquids, quantum Hall physics, and quantum information pursuits.

Fig. 1. Hoppings realized by the model Hamiltonian [Eq. (2)].

nearest-neighbor (red), next-nearest-neighbor (blue), and third-nearest-neighbor within hexagons (brown) from site i.

Fig. 2. Projected entangled simplex state.

The blue wiggly lines denote the initial state of virtual fermions c_⬡,k (blue balls) entangled across hexagons. The transparent red balls denote the projection onto the physical fermions (red balls).

Fig. 3. PEPS construction.

a Matrix product state representation of the simplex states residing on the hexagons. A can either be chosen to be of bond dimension D = 2, with $A_{12}^0=1/\sqrt{6},A_{11}^1=-A_{22}^1=1$, Q₂₁ = 1 and all other elements of A and Q equal to zero, or D = 6 with $A_{61}^0=1/\sqrt{6},A_{l,l+1}^1=1$ (l = 1, …, 5), all other elements of A equal to zero and $Q=\mathbb{1}$ (translationally invariant representation). Incoming arrows denote left and outgoing arrows right lower indices. b By combining two A tensors with the tensor M, we obtain the tensor T constituting the PEPS. c PEPS with one rank-5 tensor located on each site of the kagome lattice (gray dashed lines).

Fig. 4. Construction of the tensors T′ forming the building blocks of the interacting ψPEPS′.

The R-tensors get absorbed into the T tensor, increasing its bond dimension (indicated by thick directed lines).

Fig. 5. Entanglement spectra as a function of the many-body momentum K for different values of α for L_y = 6.

For small values of α, the low-lying spectrum strongly resembles the non-interacting one (α = 0). In particular, a cusp at K = 0 (highlighted by a red circle) is preserved as α is increased. Parallel to this, new entanglement energies appear at the top of the spectrum and eventually merge with its low-lying part.

Fig. 6. Entanglement, one-body correlation, and physical spectra.

a One-body correlation spectrum Λ_i on a thin torus with L_y = 6, b Many-body entanglement spectrum ϵ_i on an L_y = 6 torus. The red marker at k = 0 and ϵ_i = 0 is the ground state of the partition `A'. c and d are the one-body correlation spectrum Λ_i and many-body entanglement spectrum ϵ_i for L_y = 12, respectively, to clarify the variation along k. e The physical spectrum E on a cylinder of size L_x = 120 and L_y = 12 does not show an edge state with a spectral flow between the flat valence bands and dispersive conduction band.