Composite subsystem symmetries and decoration of sub-dimensional excitations

To referee 1

We want to thank the referee for the helpful report. In response to the comments in the report, we have:

1. added the definition of RD (remote detection) on page 9 when it is first used

2. replaced "Model B" with "Model FP2", consistent with other parts of the paper

3. Regarding the extension to fractal symmetries, we have updated the last paragraph in the summary section as follows:

Throughout this work, we considered planar subsystem symmetries. An interesting future direction involves extending this paper's analysis to fractal symmetries and a combination of fractal and planar symmetries. A potential issue arises in obtaining geometrically local fracton models when gauging a combination of fractal and planar symmetries. In particular, it is not obvious how to choose composite subsystem symmetries such that the set of symmetric operators is generated by purely local terms. For all (subsystem) symmetries we consider in this paper, their symmetric operators can be locally generated. For example, in a 2D system with a global Z2 symmetry, the set of symmetric operators can be generated by an on-site transverse field term and a nearest-neighbor Ising coupling term. The gauge field DOF's are then placed at the location of the Ising terms. If, on the other hand, the symmetric operators are not locally generated, the gauging process would yield a non-local model. For example, consider a 2D system with line symmetries in both the x and y directions and try gauging the composite symmetry of the tensor product of symmetries on row i and column i. In an NxN system, there are N symmetry generators. All two-body Ising coupling terms between sites (j,k) and (k,j) are now symmetric. However, since this set of terms cannot be locally generated, the gauged model would be non-local. On the other hand, gauging such composite symmetries can yield interesting k-local Hamiltonians corresponding to quantum low-density parity check (LDPC) codes. In general, obtaining LDPC codes via gauging can be insightful in understanding the associated code properties from the perspective of the gauging duality. In order to obtain an LDPC code Hamiltonian, we would want, in the ungauged model, each qubit and each relation to support and involve constant number of k-local terms respectively. Whether this is possible or not would depend on the map from initial to final ungauged symmetric terms obtained after combining the symmetries.

To referee 2

We want to thank the referee for the interesting comments. Here are our responses:

1. The idea of `twisted gauging' is an interesting one which leads to interesting observations regarding 1+1D theories. However, in this paper, we mainly consider systems with planar symmetries, which upon gauging, turn into systems with long-range entanglement. Because of this, we cannot 'conjugate' the gauging process with an SPT entangler because the systems before and after the gauging have different (generalized) symmetries.

2. Our conclusion in this paper applies to trivial and symmetry-protected paramagnets alike. In particular, the mobility of the gauge charges does not depend on whether the starting point is a trivial or non-trivial SPT.

3. If we gauge a non-anomalous subgroup of an anomalous symmetry, a simple case is that the gauged theory breaks the remaining part of the anomalous symmetry. It is not obvious to us how groupoid or noninvertible symmetries can arise. That is definitely an interesting question.

4. The interpretation of remote detectability in terms of 't Hooft anomalies is an interesting point. The principle should be similarly applicable to subsystem symmetries, although we will need to make use of a more sophisticated version of field theory as discussed in for example arXiv:2004.00015, arXiv:2008.03852, arXiv:2112.05726.

5. Thanks for pointing out the ambiguity in our terminology. We have modified the introduction section to make clear the distinction between symmetry charges and gauge charges. In particular, in the second paragraph of the Introduction, we discuss the coupling of 2 planar Z2 gauge theories. Here we clarify that forming a composite symmetry between the two Z2 symmetry generators makes the composite of the two symmetry charges no longer a symmetry charge. Additionally, we clarify that the gauge charge pair is condensed after gauging the composite Z2 symmetry.

The list of changes is as follows:

1. added the definition of RD (remote detection) on page 9 when it is first used

2. replaced "Model B" with "Model FP2", consistent with other parts of the paper

3. Regarding the extension to fractal symmetries, we have updated the last paragraph in the summary section as follows:

Throughout this work, we considered planar subsystem symmetries. An interesting future direction involves extending this paper's analysis to fractal symmetries and a combination of fractal and planar symmetries. A potential issue arises in obtaining geometrically local fracton models when gauging a combination of fractal and planar symmetries. In particular, it is not obvious how to choose composite subsystem symmetries such that the set of symmetric operators is generated by purely local terms. For all (subsystem) symmetries we consider in this paper, their symmetric operators can be locally generated. For example, in a 2D system with a global Z2 symmetry, the set of symmetric operators can be generated by an on-site transverse field term and a nearest-neighbor Ising coupling term. The gauge field DOF's are then placed at the location of the Ising terms. If, on the other hand, the symmetric operators are not locally generated, the gauging process would yield a non-local model. For example, consider a 2D system with line symmetries in both the x and y directions and try gauging the composite symmetry of the tensor product of symmetries on row i and column i. In an NxN system, there are N symmetry generators. All two-body Ising coupling terms between sites (j,k) and (k,j) are now symmetric. However, since this set of terms cannot be locally generated, the gauged model would be non-local. On the other hand, gauging such composite symmetries can yield interesting k-local Hamiltonians corresponding to quantum low-density parity check (LDPC) codes. In general, obtaining LDPC codes via gauging can be insightful in understanding the associated code properties from the perspective of the gauging duality. In order to obtain an LDPC code Hamiltonian, we would want, in the ungauged model, each qubit and each relation to support and involve constant number of k-local terms respectively. Whether this is possible or not would depend on the map from initial to final ungauged symmetric terms obtained after combining the symmetries.

4. We have modified the introduction section to make clear the distinction between symmetry charges and gauge charges. In particular, in the second paragraph of the Introduction, we discuss the coupling of 2 planar Z2 gauge theories. Here we clarify that forming a composite symmetry between the two Z2 symmetry generators makes the composite of the two symmetry charges no longer a symmetry charge. Additionally, we clarify that the gauge charge pair is condensed after gauging the composite Z2 symmetry.