Emergent anomalous higher symmetries from topological order and from dynamical electromagnetic field in condensed matter systems

Global symmetry (0-symmetry) acts on the whole space while higher $k$-symmetry acts on all the codimension-$k$ closed subspaces. The usual condensed matter lattice theories do not include dynamical electromagnetic (EM) field and do not have higher symmetries (unless we engineer fine-tuned toy models). However, for gapped systems, (anomalous) higher symmetries can emerge from the usual condensed matter theories at low energies (usually in a spontaneously broken form). We pointed out that the emergent spontaneously broken higher symmetries are nothing but a kind of topological order. Thus the study of emergent spontaneously broken higher symmetries is a study of topological order. The emergent (anomalous) higher symmetries can be used to constrain possible phase transitions and possible phases induced by certain types of excitations in topological orders. (Anomalous) higher symmetry can also emerge in gapless systems if the gapless excitations contain gapless gauge fields. In particular, EM condensed matter systems that include the dynamical EM field have an emergent anomalous $U\left(1\right)$-1-symmetry below the energy gap of the magnetic monopoles. So EM condensed matter systems can realize some physical phenomena of anomalous higher symmetry. In particular, any gapped liquid phase of an EM condensed matter system (induced by arbitrary fluctuations and condensations of electric charges and photons) must have a nontrivial bosonic topological order.

Figure 1

A cubic lattice (black) and its dual cubic lattice (red). The index $\mathit{i}$ labels the faces of the cubic lattice and the links of the dual cubic lattice.

Figure 2

A qubit model, where qubits live on the links of a honeycomb lattice. $\mathit{I}$ labels the vertices and $\mathit{i}$ labels the links (the qubits) of the honeycomb lattice.

Figure 3

The string operators.

Figure 4

The black lines describe a two-dimensional spacetime complex ${\mathcal{M}}^2$. The red lines describe the dual complex ${\tilde{\mathcal{M}}}^2$.

Figure 5

Two branched simplices with opposite orientations. (a) A branched simplex with positive orientation and (b) a branched simplex with negative orientation.

Figure 6

A 1-cochain $a$ has a value 1 on the red links: $a_{ik}=a_{jk}=1$ and a value 0 on other links: $a_{ij}=a_{kl}=0$. $da$ is nonzero on the shaded triangles: ${\left(da\right)}_{jkl}=a_{jk}+a_{kl}-a_{jl}$. For such 1 cochain, we also have $a⌣a=0$. So when viewed as a ${\mathbb{Z}}_2$-valued cochain, ${\beta }_{2}a\ne a⌣a$ mod 2.

Figure 7

A 1-cochain $a$ has a value 1 on the red links, Another 1-cochain $a^{\prime }$ has a value 1 on the blue links. On the left, $a⌣a^{\prime }$ is nonzero on the shade triangles: ${(a⌣a^{\prime })}_{ijl}=a_{ij}{a}_{jl}^{\prime }=1$. On the right, $a^{\prime }⌣a$ is zero on every triangle. Thus $a⌣a^{\prime }+a^{\prime }⌣a$ is not a coboundary.

Figure 8

Each time step of evolution is given by the path integral on a particular form of branched graph. Here is an example in $1+1\mathrm{D}$.