Anomaly Manifestation of Lieb-Schultz-Mattis Theorem and Topological Phases

The Lieb-Schultz-Mattis (LSM) theorem dictates that emergent low-energy states from a lattice model cannot be a trivial symmetric insulator if the filling per unit cell is not integral and if the lattice translation symmetry and particle number conservation are strictly imposed. In this paper, we compare the one-dimensional gapless states enforced by the LSM theorem and the boundaries of one-higher dimensional strong symmetry-protected topological (SPT) phases from the perspective of quantum anomalies. We first note that, they can be both described by the same low-energy effective field theory with the same effective symmetry realizations on low-energy modes, wherein non-on-site lattice translation symmetry is encoded as if it is a local symmetry. In spite of the identical form of the low-energy effective field theories, we show that the quantum anomalies of the theories play different roles in the two systems. In particular, We find that the chiral anomaly is equivalent to the LSM theorem, whereas there is another anomaly, which is not related to the LSM theorem but is intrinsic to the SPT states. As an application, we extend the conventional LSM theorem to multiple-charge multiple-species problems and construct several exotic symmetric insulators. We also find that the (3+1)d chiral anomaly provides only the perturbative stability of the gapless-ness local in the parameter space.Comment: 14 + 3 pages, 1 figure. (The first two authors contributed equally to the work.

Relating non-Hermitian and Hermitian quantum systems at criticality

We demonstrate three types of transformations that relate Hermitian and non-Hermitian quantum systems at criticality which can be described by conformal field theories (CFTs). For the transformation preserving both the physical spectrum (PS) and the entanglement spectrum (ES), the corresponding central charges extracted from the logarithmic scaling of the entanglement entropy are identical for both Hermitian and non-Hermitian systems. The second transformation preserves the PS but not the ES. The entanglement entropy scalings are different and lead to different central charges. We demonstrate this transformation by the dilation method for the free fermion cases, where the non-Hermitian system with central charge $c=-2$ can be mapped to the Hermitian system with $c=1$. Lastly, we study the Galois conjugation in the Fibonacci model with parameter $\phi \to -1/\phi$, in which the transformation does not preserve both PS and ES. We demonstrate the Fibonacci model and its Galois conjugation relate the tricritical Ising model/3-state Potts model and the Lee-Yang model with negative central charges from the scaling property of the entanglement entropy.Comment: 6 pages, 4 figures, comments are welcom

Symmetry-protected topological phases and quantum anomalies

We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states

Effective field theories of topological crystalline insulators and topological crystals

We present a general approach to obtain effective field theories for topological crystalline insulators whose low-energy theories are described by massive Dirac fermions. We show that these phases are characterized by the responses to spatially dependent mass parameters with interfaces. These mass interfaces implement the dimensional reduction procedure such that the state of interest is smoothly deformed into a topological crystal, which serves as a representative state of a phase in the general classification. Effective field theories are obtained by integrating out the massive Dirac fermions, and various quantized topological terms are uncovered. Our approach can be generalized to other crystalline symmetry protected topological phases and provides a general strategy to derive effective field theories for such crystalline topological phases.Comment: 20 pages, 10 figures, 1 table. Published version with minor change