Long-range order and symmetry breaking in projected entangled-pair state models

Abstract

Projected Entangled Pair States (PEPS) provide a framework for the construction of models where a single tensor gives rise to both Hamiltonian and ground state wavefunction on the same footing. A key problem is to characterize the behavior which emerges in the system in terms of the properties of the tensor, and thus of the Hamiltonian. In this paper, we consider PEPS models with Z2 onsite symmetry and study the occurence of long-range order and spontaneous symmetry breaking. We show how long-range order is connected to a degeneracy in the spectrum of the PEPS transfer operator, and how the latter gives rise to spontaneous symmetry breaking under perturbations. We provide a succinct characterization of the symmetry broken states in terms of the PEPS tensor, and find that using the symmetry broken states we can derive a local entanglement Hamiltonian, thereby restoring locality of the entanglement Hamiltonian for all gapped phases.

Holographic encoding of universality in corner spectra

“…20. This was studied previously in finite systems on a cylinder [50]. We observe that, first, there is a unique lowest entanglement eigenvalue (or one unique largest eigenvalue of corresponding transfer matrix), which is clearly identified by our method.…”

“…In Ref. [50] it was shown that there is a second-order quantum phase transition from ordered phase to disorder phase occurring at θ c ≈ 0.349596. To illustrate that our method is not limited by the usage of corner tensors, we include results from the 2d Ising PEPS in the disorder phase with θ = 0.5 in Fig.…”

“…We can obtain this entanglement spectrum using the 2d quantum state renormalization approach described earlier using CTs. In this section, we first consider the so-called Ising PEPS [50] which, by construction, has a quantum phase transition that corresponds to the classical Ising transition, which was studied earlier in Sec. V using the rCTM method.…”

“…The disorder phase: the Ising PEPS Let us first consider the Ising PEPS [50] on the square lattice with tensor A = |0 θ, θ, θ, θ|+|1 θ ,θ,θ,θ|, where the ket (bra) corresponds to the physical (virtual) degrees of freedom, and |θ = cos θ|0 + sin θ|1 as well as |θ = sin θ|0 + cos θ|1 with θ ∈ [0, π/4]. A corresponding local Hamiltonian can be written down that has this PEPS as a ground state (not shown here) [50]. In Ref.…”

Classifying parafermionic gapped phases using matrix product states

“…The framework of MPS and PEPS naturally provides an efficient method to classify topologically ordered phases 17,18 , symmetry protected topological phases [19][20][21] , and the long-range ordered phases with spontaneous symmetry breaking 19,22 . However, it is not straight forward to extend the MPS representation to the one-dimensional parafermion systems.…”