Topological crystalline insulators (TCIs) are nontrivial quantum phases of matter protected by crystalline (and other) symmetries. They are originally predicted by band theories, so an important question is their stability under interactions. In this paper, by directly studying the physical bulk properties of several band-theory-based nontrivial TCIs that are conceptually interesting and/or experimentally feasible, we show they are stable under interactions. These TCIs include (1) a weak topological insulator, (2) a TCI with a mirror symmetry and its time-reversal symmetric generalizations, (3) a doubled topological insulator with a mirror symmetry, and (4) two TCIs with symmetry-enforced-gapless surfaces. We describe two complementary methods that allow us to determine the properties of the magnetic monopoles obtained by coupling these TCIs to a U(1) gauge field. These methods involve studying different types of surface states of these TCIs. Applying these methods to our examples, we find all of them have nontrivial monopoles, which proves their stability under interactions. Furthermore, we discuss two levels of relations between these TCIs and symmetry enriched U(1) quantum spin liquids (QSLs). First, these TCIs are directly related to U(1) QSLs with crystalline symmetries. Second, there is an interesting correspondence between U(1) QSLs with crystalline symmetries and U(1) QSLs with internal symmetries. In particular, the TCIs with symmetry-enforced-gapless surfaces are related to the “fractional topological paramagnets” introduced in Zou et al. [arXiv:1710.00743].
An ultimate theory of a phase of matter should describe all its universal properties via quantities that are measurable numerically and experimentally. In this work, we present a microscopic universal theory of symmetry-enriched topological quantum spin liquids (TQSLs) in two spatial dimensions, which directly utilizes microscopically measurable quantities to describe the universal properties. This theory applies to generic TQSLs, which can be Abelian or non-Abelian, chiral or non-chiral. The symmetries are also general, which can include both internal and lattice symmetries, unitary and anti-unitary symmetries, and discrete and continuous symmetries. There can be spin-orbit coupling, the microscopic degrees of freedom may transform linearly or projectively under the symmetries, and the symmetries can permute anyons. The input of the theory is some microscopic states with anyons, operators that control the dynamics of anyons, and symmetry actions in the TQSL, and its output is a set of data characterizing the universal properties, whose underlying mathematical structure is a generalization of category theory. Based on this theory, we find an explicit bijective map between the universal data characterizing a TQSL with a symmetry described by a group
G, where the symmetry actions may include both lattice and internal symmetries, and the corresponding universal data for a TQSL with only an internal symmetry group
G, and thus establish a precise crystalline equivalence principle. We demonstrate our theory in symmetry-enriched TQSLs realized on quantum processors based on superconducting qubits, trapped ions, and Rydberg atoms, and in each example we verify the Lieb-Schultz-Mattis anomaly matching condition. Our theory provides a solid basis for identifying and manipulating symmetry-enriched TQSLs, which further paves the way for fault-tolerant quantum computation based on these systems.