Detecting subsystem symmetry protected topological order via entanglement entropy

Abstract

Subsystem symmetry-protected topological (SSPT) order is a type of quantum order that is protected by symmetries acting on lower-dimensional subsystems of the entire system. In this paper, we show how SSPT order can be characterized and detected by a constant correction to the entanglement area law, similar to the topological entanglement entropy. Focusing on the paradigmatic two-dimensional cluster phase as an example, we use tensor network methods to give an analytic argument that almost all states in the phase exhibit the same correction to the area law, such that this correction may be used to reliably detect the SSPT order of the cluster phase. Based on this idea, we formulate a numerical method that uses tensor networks to extract this correction from ground state wave functions. We use this method to study the fate of the SSPT order of the cluster state under various external fields and interactions, and find that the correction persists unless a phase transition is crossed, or the subsystem symmetry is explicitly broken. Surprisingly, these results uncover that the SSPT order of the cluster state persists beyond the cluster phase, thanks to a new type of subsystem time-reversal symmetry. Finally, we discuss the correction to the area law found in 3D cluster states on different lattices, indicating rich behaviour for general subsystem symmetries.The modern perspective of quantum phases of matter is based on global patterns of entanglement [1][2][3]. Two quantum states are said to lie in distinct (symmetryprotected) topological phases when they cannot be connected by (symmetric) local unitary evolution. As such, topological phases of matter can be characterized and detected by entanglement-based quantities. A well-known example is the topological entanglement entropy (TEE): For states with non-trivial topological order, the scaling of entanglement entropy exhibits a correction to the area law [4][5][6][7]. That is, for a subregion A of a lattice, the entropy for ground states of gapped, local Hamiltonians takes the general form,for some constants a, γ, where ∂A is the boundary of region A and the dots indicate terms that decay exponentially with |A|. The correction γ takes a uniform non-zero value within non-trivial topological phases as a consequence of the global entanglement patterns. It can therefore be used to detect and characterize topological order and topological phase transitions analytically, numerically, and potentially even experimentally [8][9][10][11][12][13][14][15][16][17].Recently, however, it has been observed that γ may deviate from the expected value due to the presence of symmetry-protected topological (SPT) order localized around ∂A [18-23]. One setting in which this occurs is for states with subsystem SPT (SSPT) order [21,[24][25][26]. Such order is non-trivial only in the presence of subsystem symmetries, which are defined as symmetries that act on rigid lower-dimensional subsystems of the entire system. In cases where these symmetries act on 1D lines spanning a 2D lattice, one may ...

Subsystem symmetry enriched topological order in three dimensions

“…The TEE refers to a constant correction to the area law when computing bipartite entanglement entropy, and is a topological invariant, in the sense that it takes a uniform value within a given topological phase of matter [49,50]. Recently, it has been understood that there are certain quantum states for which the TEE, for certain bipartitions, does not match the expected value [18,20,22,[51][52][53][54][55]. In a majority of known cases (see Ref.…”

“…In fact, it was shown in Ref. [22] that this "spurious" TEE takes a uniform value within the SSPT phase of the 2D cluster state, and can be used to detect SSPT order and phase transitions. To emphasize its relation to the SSPT order, it was dubbed the symmetry-protected entanglement entropy (SPEE).…”

“…In this section, we show that the topological entanglement entropy (TEE) of our 3D SSET state is larger than that of the 3D toric code when the boundaries of bipartitions are aligned with the symmetry planes. This is in analogy to the fact that states with 2D SSPT order, such as the cluster state, have a nonzero TEE when bipartitions are aligned with the linelike symmetries, despite the absence of topological order [18,22,52,54]. To emphasize that this value comes from the SSPT order, rather than topological order, we call it the symmetry-protected entanglement entropy (SPEE) [22].…”

“…A new paradigm in the classification of gapped phases of matter has recently begun thanks to the discovery of models with novel subdimensional physics. This includes the fracton topological phases [1][2][3][4][5][6][7][8][9][10][11][12], in which topological quasiparticles are either immobile or confined to move only within subsystem such as lines or planes, as well as models with subsystem symmetries, which are symmetries that act nontrivially only on rigid subsystems of the entire system [7,8,[13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. These two types of models are dual since gauging subsystem symmetries can result in fracton topological order, analogous to how topological order can be obtained by gauging a global symmetry [7,8,15].…”

“…Models with subsystem symmetries are also interesting in their own right. For example, one may use them to define symmetry-protected topological (SPT) phases, giving rise to the so-called subsystem SPT (SSPT) phases [16][17][18][19][20][21][22]26,28]. SSPT phases display new physics compared to conventional SPT phases such as extensive edge degeneracy and unique entanglement properties.…”