Quantum Anomaly Manifestation By Dimensional Reduction

by Tommy on 12/05/2017

This is how the quantum cosmology dark matter axion nutshell will be ultimately cracked.


Anomaly Manifestation of Lieb-Schultz-Mattis Theorem and Topological Phases, Gil Young Cho and Shinsei Ryu (10 May 2017)

The Lieb-Schultz-Mattis theorem constrains the possible low-energy and long-distance behaviors of states which emerge from microscopic lattice Hamiltonians. The theorem dictates that the emergent state cannot be a trivial symmetric insulator if the filling per unit cell is not integral and if lattice translation symmetry and particle number conservation are strictly imposed. Investigating one-dimensional symmetric gapless states which are forced to be critical by the theorem, we show that the theorem, the absence of a trivial insulator phase at non-integral filling, has a very close connection to quantum anomaly. We further show that, in terms of symmetry realizations on low-energy modes, low-energy spectrum, and anomaly, the gapless states emergent from lattice Hamiltonians are equivalent to the boundary theory of the strong symmetry-protected topological phases in one-higher dimensions, where non-local translational symmetry of the lattice is encoded as some local symmetry. Once a global symmetry is realized in a non-on-site fashion, the boundary of the topological phases can be realized in a stand-alone lattice model, and the no-go theorem for the boundary is circumvented, similar to the recent discussions of the half-filled Landau level and topological insulators. Finally we extend our analysis to the higher-dimensional example, the Dirac semimetal in three spatial dimensions.

See also: https://arxiv.org/abs/1705.00012

Lieb-Schultz-Mattis Theorem and its generalizations from the Perspective of the Symmetry Protected Topological phase, Chao-Ming Jian, Zhen Bi and Cenke Xu (28 April 2017)

We ask whether a local Hamiltonian with a featureless (fully gapped and nondegenerate) ground state could exist in certain quantum spin systems. We address this question by mapping the vicinity of certain quantum critical point (or gapless phase) of the d−dimensional spin system under study to the boundary of a (d+1)−dimensional bulk state, and the lattice symmetry of the spin system acts as an on-site symmetry in the field theory that describes both the selected critical point of the spin system, and the corresponding boundary state of the (d+1)−dimensional bulk. If the symmetry action of the field theory is non-anomalous, i.e. the corresponding bulk state is a trivial state instead of a bosonic symmetry protected topological (SPT) state, then a featureless ground state of the spin system is allowed; if the corresponding bulk state is indeed a nontrivial SPT state, then it likely excludes the existence of a featureless ground state of the spin system. From this perspective we identify the spin systems with SU(N) and SO(N) symmetries on one, two and three dimensional lattices that permit a featureless ground state. We also verify our conclusions by other methods, including an explicit construction of these featureless spin states.

These are long and complicated papers with lots of references, so take your time.

Nevertheless, if you persist, you will get up to speed on this.


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