ICQM Faculty Prof. Xiong-Jun Liu et al predict the existence of unconventional type-II Nambu-Goldstone bosons with topological origin

ICQM faculty Prof. Xiong-Jun Liu, together with Dr. Jian-Song Pan, a postdoc in Wilczek Quantum Center and visiting postdoc of Xiong-Jun Liu’s group, and Prof. Vincent Liu at University of Pittsburgh published a paper in Physical Review Letters, predicting in s-p band superfluids the existence of unconventional type-II Nambu-Goldstone bosons with topological origin. In particular, they found that this prediction intrinsically bridges the understanding of quantum matter through Landau symmetry-breaking paradigm and topological theory. The paper is published as J.-S. Pan, V. W. Liu, and X.-J. Liu, Phys. Rev. Lett. 125, 260402 (2020); The link to the paper: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.125.260402

Characterizing quantum phases within a unified framework is the basic pursuit of condensed matter physics. The most celebrated theories include the Landau's symmetry-breaking paradigm and the topological theory. In the former the phase transition is associated with the spontaneous symmetry breaking and emergence of local order parameter, with the low-energy physics being captured by the emergent Nambu-Goldstone (NG) modes. In the latter, instead of being characterized by symmetry breaking and local order, the topological phase transition is characterized by the change of global topology. The nontrivial topological phases, e.g. with bulk gap host gapless boundary modes which are linked to the bulk topology. It has been typically considered that the symmetry-breaking paradigm and the topological theory are mutually exclusive, namely, they characterize conceptually different phases.

The prediction of the unconventional Nambu-Goldstone bosons by Xiong-Jun Liu and collaborators brings about a fundamentally new paradigm which intrinsically bridges the symmetry-breaking and topological theories. The NG modes in non-relativistic system are classified into two types: being of either an odd (type I) or an even (type II) power energy-momentum dispersion. Unlike the type-I gapless NG modes which appear generally when a continuous symmetry is broken spontaneously, the current broadly developed theory indicates that the emergence of the type-II NG modes requires a pair of noncommutative symmetries to be simultaneously broken. In the published letter, the predicted unconventional type-II NG modes cannot be interpreted with the celebrated symmetry-based argument, but relates to an emergent topological transition. The broken continuous symmetries in the proposed superfluids commute. Thus according to the conventional symmetry-based theory, no type-II NG modes should be expected. Surprisingly, the authors show that a branch of type-II NG modes emerges at a phase-diagram boundary, across which the degenerate space of ground states, generated by the broken symmetries, undergoes an emergent topological transition after being projected onto the complex order-parameter subspace, with a a generic framework being established.

The unconventional type-II NG modes predicted here intrinsically combines the two most fundamental and broad theories in condensed matter physics: the Landau symmetry-breaking theory and topological theory. This is also different from the famous issue “topological superconductor/topological superfluid”, which does not intrinsically combine the Landau symmetry-breaking and topological theory, since in topological superconductor (topological superfluid) the symmetry-breaking phase (superconductor) and topology are defined independently. In sharp contrast, in the present prediction, the topological transition is defined in the projection space of the broken symmetries, leading to the first paradigm combining the Landau symmetry-breaking and topological theories intrinsically. On the other hand, the authors also proposed a feasible scheme based on s-p band Bose superfluids to realize and observe the predicted unconventional type-II NG modes. As a result, this work may potentially have significant impact on both the theoretical and the experimental studies of quantum phases, and may bring about a new category framework for quantum matters.

Dr. Jian-Song Pan is the first author of the paper, Prof. Xiong-Jun Liu and Prof. Vincent Liu are the corresponding authors of the paper. This work was supported by the fundings from NSFC, MOST, CAS, etc.

(Upper) (a) Phase diagram of s-p orbital superfluids.（b-e）Spectra of the Nambu-Goldstone bosons. The unconventional type-II Nambu-Goldstone bosons are obtained at the phase boundary between M and T phases in (a).（Lower）For the transition from M phase to T phase, the symmetry-breaking is unchanged, but an emergent topological transition in the projected order-parameter space is obtained at the phase boundary, which leads to the emergence of the unconventional type-II Nambu-Goldstone modes.