Flux-Fusion Anomaly Test and Bosonic Topological Crystalline Insulators

We introduce a method, dubbed the flux-fusion anomaly test, to detect certain anomalous symmetry fractionalization patterns in two-dimensional symmetry-enriched topological (SET) phases. We focus on bosonic systems with ${\mathbb{Z}}_2$ topological order and a symmetry group of the form $G=\mathrm{U}(1)⋊G^{\prime }$, where $G^{\prime }$ is an arbitrary group that may include spatial symmetries and/or time reversal. The anomalous fractionalization patterns we identify cannot occur in strictly $d=2$ systems but can occur at surfaces of $d=3$ symmetry-protected topological (SPT) phases. This observation leads to examples of $d=3$ bosonic topological crystalline insulators (TCIs) that, to our knowledge, have not previously been identified. In some cases, these $d=3$ bosonic TCIs can have an anomalous superfluid at the surface, which is characterized by nontrivial projective transformations of the superfluid vortices under symmetry. The basic idea of our anomaly test is to introduce fluxes of the U(1) symmetry and to show that some fractionalization patterns cannot be extended to a consistent action of $G^{\prime }$ symmetry on the fluxes. For some anomalies, this can be described in terms of dimensional reduction to $d=1$ SPT phases. We apply our method to several different symmetry groups with nontrivial anomalies, including $G=\mathrm{U}(1)\times {\mathbb{Z}}_2^T$ and $G=\mathrm{U}(1)\times {\mathbb{Z}}_2^P$, where ${\mathbb{Z}}_2^T$ and ${\mathbb{Z}}_2^P$ are time-reversal and $d=2$ reflection symmetry, respectively.

Popular Summary

One way to identify a new quantum phase of matter is to show that another quantum phase of matter is impossible. There are theories naively describing a $d$-dimensional phase of matter that cannot be realized in a $d$-dimensional system but can be realized at the surface of a ($d+1$)-dimensional system. Such theories are referred to as anomalous, and proving a theory to be anomalous provides a route to identifying previously unknown quantum phases of matter. Here, we develop an approach to show that certain two-dimensional theories are anomalous, and we identify new phases of matter in three dimensions.

There are a number of fascinating quantum phases of matter whose edge or surface states are known to be anomalous. One example is the quantum Hall effect, which occurs in two-dimensional electron systems. The anomalous one-dimensional edge includes electrons moving only in one direction, as if going down a one-way street.

So-called topological crystalline insulators are similar, except that the symmetry of rigid motions of the crystal lattice plays a key role. Here, by showing that certain two-dimensional theories are anomalous, we theoretically identify several examples of topological crystalline insulators of bosons in three dimensions.

We expect that our findings will pave the way for the discovery of new topological crystalline insulators.

Figure 1

Illustration of the operations generating the $d=2$ space group $pm$. A square lattice, which is invariant under $pm$ symmetry, is shown to aid visualization. $T_x$ and $T_y$ are translations by one lattice constant along the $x$ and $y$ axes, respectively. $P_x$ is a reflection, with the axis indicated by the left-hand vertical dashed line. The group $pm$ has two types of reflection axes, with $P_x$ being of one type and $T_x{P}_x$ being of the other type. The reflection axis for $T_x{P}_x$ is shown as the right-hand vertical dashed line.

Figure 2

Illustration of the flux-fusion anomaly test using dimensional reduction to a $d=1$ cylinder. The ${\mathbb{Z}}_2\subset \mathrm{U}(1)$ flux $\mathrm{\Omega }$ is threaded twice along the left-hand cylinder, while $m$ is threaded along the right-hand cylinder. Because of the fusion rule ${\mathrm{\Omega }}^2=m$, these two systems are in the same $d=1$ SPT phase.

Figure 3

Coupled-layer construction. Each layer is a SET phase with ${\mathbb{Z}}_2$ topological order. $E$ and $M$ particles transform nontrivially under symmetry $G$, while $e$ and $m$ transform trivially. Composite particles indicated by ovals are condensed to obtain a $d=3$ SPT phase (which may be the trivial SPT phase). The particles in dashed boxes remain deconfined and uncondensed, and give rise to surface SET phases at the top and bottom surfaces. By choosing the fractionalization classes of $E$ and $M$, surface SET phases with any desired symmetry fractionalization pattern can be realized by this construction.

Figure 4

Illustration of the operations generating the $d=2$ space group $p4mm$, the symmetry group of the square lattice. $T_x$ and $T_y$ are translations by one lattice constant along the $x$ and $y$ axes, respectively. The vertical dashed line is the axis for the reflection $P_x$, and the diagonal dashed line is the axis for the reflection $P_{xy}$.