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arXiv:2010.15124 · Physical review. B./Physical review. B

Bifurcating subsystem symmetric entanglement renormalization in two dimensions

Jonathan F. San Miguel, Arpit Dua, Dominic J. Williamson

Published Jan 29, 2021• 17 citations
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Abstract

We introduce the subsystem symmetry-preserving real-space entanglement renormalization group and apply it to study bifurcating flows generated by linear and fractal subsystem symmetry-protected topological phases in two spatial dimensions. We classify all bifurcating fixed points that are given by subsystem symmetric cluster states with two qubits per unit cell. In particular, we find that the square lattice cluster state is a quotient-bifurcating fixed point, while the cluster states derived from Yoshida's first-order fractal spin liquid models are self-bifurcating fixed points. We discuss the relevance of bifurcating subsystem symmetry-preserving renormalization group fixed points for the classification and equivalence of subsystem symmetry-protected topological phases.

DOI
10.1103/physrevb.103.035148

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We introduce the subsystem symmetry-preserving real-space entanglement renormalization group and apply it to study bifurcating flows generated by linear and fractal subsystem symmetry-protected topological phases in two spatial dimensions. We classify all bifurcating fixed points that are given by subsystem symmetric cluster states with two qubits per unit cell. In particular, we find that the square lattice cluster state is a quotient-bifurcating fixed point, while the cluster states derived from Yoshida's first-order fractal spin liquid models are self-bifurcating fixed points. We discuss the relevance of bifurcating subsystem symmetry-preserving renormalization group fixed points for the classification and equivalence of subsystem symmetry-protected topological phases.
We consider symmetry-protected topological (SPT) phases in two dimensions protected by linear subsystem symmetries, i.e., those that act along rigid lines. There is a distinction between a ``strong'' subsystem SPT phase and a ``weak'' one, which is composed of decoupled one-dimensional SPTs with global symmetries. We propose a natural definition for strong equivalence of such phases, in terms of a linearly symmetric local unitary transformation, under which a weak subsystem SPT is equivalent to the trivial phase. This leads to a number of distinct equivalence classes of strong subsystem SPTs, which we show are in one-to-one correspondence with elements of the group C[Gs]=H2[Gs2,U(1)]/H2[Gs,U(1)]3\mathcal{C}[{G}_{s}]={\mathcal{H}}^{2}[{G}_{s}^{2},U(1)]/{{{\mathcal{H}}^{2}[{G}_{s},U(1)]}}^{3}C[Gs​]=H2[Gs2​,U(1)]/H2[Gs​,U(1)]3, where Gs{G}_{s}Gs​ is the finite Abelian on-site symmetry group. We also show that strong subsystem SPTs by our classification necessarily exhibit a spurious topological entanglement entropy on a cylinder.
Entanglement renormalization is a real-space renormalization group (RG) transformation for quantum many-body systems. It generates the multiscale entanglement renormalization ansatz (MERA), a tensor network capable of efficiently describing a large class of many-body ground states, including those of systems at a quantum critical point or with topological order. The MERA has also been proposed to be a discrete realization of the holographic principle of string theory. Here we propose the use of symmetric tensors as a mechanism to build a symmetry-protected RG flow, and discuss two important applications of this construction. First, we argue that symmetry-protected entanglement renormalization produces the proper structure of RG fixed points, namely, a fixed-point for each symmetry-protected phase. Second, in the context of holography, we show that by using symmetric tensors, a global symmetry at the boundary becomes a local symmetry in the bulk, thus explicitly realizing in the MERA a characteristic feature of the AdS/CFT correspondence.
A defining feature of a symmetry protected topological phase (SPT) in one dimension is the degeneracy of the Schmidt values for any given bipartition. For the system to go through a topological phase transition separating two SPTs, the Schmidt values must either split or cross at the critical point in order to change their degeneracies. A renormalization group (RG) approach based on this splitting or crossing is proposed, through which we obtain an RG flow that identifies the topological phase transitions in the parameter space. Our approach can be implemented numerically in an efficient manner, for example, using the matrix product state formalism, since only the largest first few Schmidt values need to be calculated with sufficient accuracy. Using several concrete models, we demonstrate that the critical points and fixed points of the RG flow coincide with the maxima and minima of the entanglement entropy, respectively, and the method can serve as a numerically efficient tool to analyze interacting SPTs in the parameter space.
We show that two-dimensional fractal subsystem symmetry-protected topological phases may serve as resources for universal measurement-based quantum computation. This is demonstrated explicitly for two cluster models known to lie within fractal symmetry-protected topological phases, and computational universality is shown to persist throughout those phases. One of the models considered is simply the cluster model on the honeycomb lattice in one limit. We discuss the importance of rigid subsystem symmetries, as opposed to global or $(D\ensuremath{-}1)$-form symmetries, in this context.
We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization approach that removes local entanglement and produces a coarse-grained lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixed-point tensors Tinv{T}_{\text{inv}}Tinv​ plus the symmetry group Gsym{G}_{\text{sym}}Gsym​ of the tensors (i.e., the symmetry group of the Lagrangian) characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by two-dimensional (2D) statistical Ising model, 2D statistical loop-gas model, and 1+1D1+1\text{D}1+1D quantum spin-1/2 and spin-1 models. In particular, using such a (Gsym,Tinv)({G}_{\text{sym}},{T}_{\text{inv}})(Gsym​,Tinv​) characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry-protected topological phase. The (Gsym,Tinv)({G}_{\text{sym}},{T}_{\text{inv}})(Gsym​,Tinv​) characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.
We study symmetry-protected topological (SPT) phases of matter in 2D2\mathrm{D}2D protected by symmetries acting on fractal subsystems of a certain type. Despite the total symmetry group of such systems being subextensively large, we show that only a small number of phases are actually realizable by local Hamiltonians. Which phases are possible depends crucially on the spatial structure of the symmetries, and we show that in many cases, no nontrivial SPT phases are possible at all. In cases where nontrivial SPT phases do exist, we give an exhaustive enumeration of them in terms of their locality.