The interplay between symmetry and topology leads to a rich variety of electronic topological phases, protecting states such as the topological insulators and Dirac semimetals. Previous results, like the Fu-Kane parity criterion for inversion-symmetric topological insulators, demonstrate that symmetry labels can sometimes unambiguously indicate underlying band topology. Here we develop a systematic approach to expose all such symmetry-based indicators of band topology in all the 230 space groups. This is achieved by first developing an efficient way to represent band structures in terms of elementary basis states, and then isolating the topological ones by removing the subset of atomic insulators, defined by the existence of localized symmetric Wannier functions. Aside from encompassing all earlier results on such indicators, including in particular the notion of filling-enforced quantum band insulators, our theory identifies symmetry settings with previously hidden forms of band topology, and can be applied to the search for topological materials.

In analogy with crystalline solids around us, Wilczek recently proposed the idea of "time crystals" as phases that spontaneously break the continuous time translation into a discrete subgroup. The proposal stimulated further studies and vigorous debates whether it can be realized in a physical system. However, a precise definition of the time crystal is needed to resolve the issue. Here we first present a definition of time crystals based on the time-dependent correlation functions of the order parameter. We then prove a no-go theorem that rules out the possibility of time crystals defined as such, in the ground state or in the canonical ensemble of a general Hamiltonian, which consists of not-too-long-range interactions.Recently, Wilczek proposed a fascinating new concept of time crystals, which spontaneously break the continuous time translation symmetry, in analogy with ordinary crystals which break the continuous spatial translation symmetry [1][2][3]. Li et al. soon followed with a concrete proposal for an experimental realization and observation of a (space-)time crystal, using trapped ions in a ring threaded by an Aharonov-Bohm flux [4][5][6]. While the proposal of time crystals was quite bold, it is, on the other hand, rather natural from the viewpoint of relativity: since we live in the Lorentz invariant space-time, why don't we have time crystals if there are ordinary crystals with a long-range order in spatial directions?However, the very existence, even as a matter of principle, of time crystals is rather controversial. For example, Bruno [7] and Nozières [8] discussed some difficulties in realizing time crystals. However, since their arguments were not fully general, several new realizations of time crystals, which avoid these no-go arguments, were proposed [9,10].In fact, a part of the confusion can be attributed to the lack of a precise mathematical definition of time crystals. Here, we first propose a definition of time crystals in the equilibrium, which is a natural generalization of that of ordinary crystals and can be formulated precisely also for time crystals. We then prove generally the absence of time crystals defined as such, in the equilibrium with respect to an arbitrary Hamiltonian which consists of not-too-long-range interactions. We present two theorems: one applies only to the ground state, and the other applies to the equilibrium with an arbitrary temperature.Naively, time crystals would be defined in terms of the expectation value Ô (t) of an observableÔ(t). If Ô (t) exhibits a periodic time dependence, the system may be regarded as a time crystal. However, the very definition of eigenstatesĤ|n = E n |n immediately implies that the expectation value of any Heisenberg operatorÔ(t) ≡ e iĤtÔ (0)e −iĤt in the Gibbs equilibrium ensemble is time independent. To see this, recall that the expectation value X is defined as X ≡ 0|X|0 at zero temperature T = 0 and X ≡ tr(Xe −βĤ )/Z = n n|X|n e −βEn /Z at a finite temperature T = β −1 > 0, where |0 is the ground state and Z ≡ tr[e −βĤ ] is th...

The rich variety of crystalline symmetries in solids leads to a plethora of topological crystalline insulators (TCIs) featuring distinct physical properties, which are conventionally understood in terms of bulk invariants specialized to the symmetries at hand. While isolated examples of TCI have been identified and studied, the same variety demands a unified theoretical framework. In this work, we show how the surfaces of TCIs can be analyzed within a general surface theory with multiple flavors of Dirac fermions, whose mass terms transform in specific ways under crystalline symmetries. We identify global obstructions to achieving a fully gapped surface, which typically lead to gapless domain walls on suitably chosen surface geometries. We perform this analysis for all 32 point groups, and subsequently for all 230 space groups, for spin-orbit-coupled electrons. We recover all previously discussed TCIs in this symmetry class, including those with "hinge" surface states. Finally, we make connections to the bulk band topology as diagnosed through symmetry-based indicators. We show that spin-orbit-coupled band insulators with nontrivial symmetry indicators are always accompanied by surface states that must be gapless somewhere on suitably chosen surfaces. We provide an explicit mapping between symmetry indicators, which can be readily calculated, and the characteristic surface states of the resulting TCIs. arXiv:1711.11589v3 [cond-mat.str-el]

Using the effective Lagrangian approach, we clarify general issues about Nambu-Goldstone bosons without Lorentz invariance. We show how to count their number and study their dispersion relations. Their number is less than the number of broken generators when some of them form canonically conjugate pairs. The pairing occurs when the generators have a nonzero expectation value of their commutator. For non-semi-simple algebras, central extensions are possible. The underlying geometry of the coset space in general is partially symplectic.