Subsystem Symmetry Fractionalization and Foliated Field Theory

We thank the Referees for their detailed reports, we have responded to their questions and comments below. An updated manuscript with the changes outlined has been resubmitted. With these changes, we believe the updated draft is ready for publication.

Referee 1

We respond to the referee’s comments in order below:

1. We have added the following paragraph to clarify this point:

“Since the original one-form symmetry on non-contractible cycles is still nontrivial after ``gauging the contractible one-form symmetry'', the resulting theory still has anomalous $\mathbb{Z}_2\times\mathbb{Z}_2$ one-form symmetry, which guarantees nontrivial ground state degeneracy.”

1. We have added the following paragraph to clarify that due to gauging a subsystem subgroup of the 1-form symmetry of toric code, the line operators referenced do not become trivial, and so can satisfy a nontrivial global relation:

“The Gauss law in Figure 2 is different from the usual Gauss law imposed when gauging ordinary one-form symmetry (see e.g.~Refs). In the ordinary case, the Gauss law term is $\hat X_pX_e\hat X_{p'}$ for every edge $e$, and the two adjacent plaquettes $p,p'$. Such a Gauss law constraint is stronger than the one imposed in Figure 1. The latter constraint does not imply that all closed $X$-loops become trivial, while the former does. The $X$-loops that become trivial under the former Gauss law constraints are precisely those that are contractible, i.e. formed by product of the Hamiltonian star terms.”

1. We have added a comment to clarify the fixing of edge degrees of freedom:

“On an infinite planar lattice, using the Gauss law constraints we can project out and remove the edge qubits, to arrive at a new model with face qubits only.\footnote{ On a more general lattice, we cannot gauge-fix the edge qubit completely.}”

1. We thank the referee for pointing out this typo, we have corrected it in the updated submission.

2. Below eq. 2.4 we have added a comment to clarify this point:

“The quotient is generated by the symmetry with current \begin{equation} J_0:=j_0^x-j_0^y,\quad J_x:=j_x^y,\quad J_y:=-j_y^x,\quad \partial_0J_0+\partial_xJ_x+\partial_yJ_y=0~. \end{equation} In other words, by gauging the symmetry we impose the global relation. We remark that since the relation demands that product of lines in $x$ directions to be equal to product of lines in $y$ directions, while individually a line is not contractable, the product of suitable $x,y$ lines are contractible, and they can be expressed as product of small contractible loops. “

1. Below eq. 2.6 we have added clarification:

“where $\lambda^x=\lambda^x(t,y)$ and $\lambda^y=\lambda^y(t,x)$ to the maintain the vanishing components $A^x_x=0$ and $A^y_y=0$ that do not couple to the current.”

1. We thank the referee for pointing out these typos, they have been corrected in the updated submission.

Referee 2

We respond to the referee’s comments below.

The use of the terminology “subsystem one-form symmetry” (e.g. around equation 1.2) is somewhat confusing on first read. The authors seem to mean “0-form subsystem symmetry which is a subgroup of an ordinary one-form symmetry” but the language used might leave readers confusing it with what is sometimes called a “one-form subsystem symmetry” which is, for example, the kind of symmetry that the X-cube model has. I think the authors can be more careful and consistent with their language throughout.

We have added the following paragraph above the first use of “subsystem one-form symmetry” to clarify the terminology:

“We use the following terminology throughout this work: a global symmetry, as characterized by a generator that commutes with the Hamiltonian, is called a $q$-form symmetry whenever the generator has support on a codimension-$q$ subspace in space. A symmetry is called a subsystem symmetry if the symmetry generator is not fully topological, i.e. the eigenvalue of the generator changes if we deform its support in general directions, even when it is away from other operators. These adjectives apply to global symmetries, and when a symmetry obeys the above two properties, we call it a subsystem $q$-form symmetry.”

This paragraph is regarding the toric code example analyzed in Section 1.1.1. The standard gauging of the Z2 one-form symmetry of the 2+1d toric code looks very different from what is carried out in the text. For example, in the standard approach, the end result of the gauging is essentially the 2+1d transverse field Ising model on the dual lattice. The difference between these two gauging procedures appears to stem from the Gauss law terms used in Figure 1, which differ from the more conventional choice XpXeXp′, where e is an edge and p and p′ are the two plaquettes which share e on their boundaries. Can the authors comment on the relationship between their approach and the more standard approach? Perhaps the difference is due to the fact that the authors say they are gauging the “contractible part of the one-form symmetry”, but it is not entirely clear to me what gauging a contractible one-form symmetry means in general.

We have added a paragraph to clarify our procedure to gauge the subsystem symmetry subgroup of the one-form symmetry of the 2+1d toric code:

“The Gauss law in Figure 2 is different from the usual Gauss law imposed when gauging ordinary one-form symmetry (see e.g.~Refs). In the ordinary case, the Gauss law term is $\hat X_pX_e\hat X_{p'}$ for every edge $e$, and the two adjacent plaquettes $p,p'$. Such a Gauss law constraint is stronger than the one imposed in Figure 1. The latter constraint does not imply that all closed $X$-loops become trivial, while the former does. The $X$-loops that become trivial under the former Gauss law constraints are precisely those that are contractible, i.e. formed by product of the Hamiltonian star terms.”

Some small typos/comments

We thank the referee for pointing out these typos, they have been corrected in the updated submission.