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arXiv:2002.10479 · Physical review. B./Physical review. B

Topological edge and interface states at bulk disorder-to-order quantum critical points

Yichen Xu, Xiao-Chuan Wu, Chao‐Ming Jian, Cenke Xu

Published May 18, 2020• 14 citations
arXivHTMLPDF

Abstract

We study the interplay between two nontrivial boundary effects: (1) the two-dimensional ($2d$) edge states of three-dimensional ($3d$) strongly interacting bosonic symmetry-protected topological states, and (2) the boundary fluctuations of $3d$ bulk disorder-to-order phase transitions. We then generalize our study to $2d$ gapless states localized at an interface embedded in a $3d$ bulk, when the bulk undergoes a quantum phase transition. Our study is based on generic long-wavelength descriptions of these systems and controlled analytic calculations. Our results are summarized as follows: (i) The edge state of a prototype bosonic symmetry-protected state can be driven to a new fixed point by coupling to the boundary fluctuations of a bulk quantum phase transition; (ii) the states localized at a $2d$ interface of a $3d\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N)$ quantum antiferromagnet may be driven to a new fixed point by coupling to the bulk quantum critical modes. Properties of the new fixed points identified are also studied.

DOI
10.1103/physrevb.101.184419

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We study the interplay between two nontrivial boundary effects: (1) the two-dimensional (2d2d2d) edge states of three-dimensional (3d3d3d) strongly interacting bosonic symmetry-protected topological states, and (2) the boundary fluctuations of 3d3d3d bulk disorder-to-order phase transitions. We then generalize our study to 2d2d2d gapless states localized at an interface embedded in a 3d3d3d bulk, when the bulk undergoes a quantum phase transition. Our study is based on generic long-wavelength descriptions of these systems and controlled analytic calculations. Our results are summarized as follows: (i) The edge state of a prototype bosonic symmetry-protected state can be driven to a new fixed point by coupling to the boundary fluctuations of a bulk quantum phase transition; (ii) the states localized at a 2d2d2d interface of a 3dSU(N)3d\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N)3dSU(N) quantum antiferromagnet may be driven to a new fixed point by coupling to the bulk quantum critical modes. Properties of the new fixed points identified are also studied.
Bulk topology and criticality can both lead to nontrivial boundary effects. Topological orders are often characterized by their robust edge states, while bulk critical points can have different boundary scalings governed by boundary conditions. The interplay between these two different boundary effects is an intriguing problem. The boundary of the deconfined quantum critical point (DQCP) is the ideal platform for the interplay of the two boundary effects, as the DQCP is also an intrinsically gapless symmetry protected topological (igSPT) state. In this work we discuss the boundary of several analogues of the DQCP. We demonstrate that the fluctuation of the bulk order parameters and their various boundary conditions lead to a rich possibility of the edge states, including a "pseudogap" (or super power-law decay) behavior. We also discuss the quantum information perspective of our work, i.e. the implication of our results on DQCP under weak-measurement. Weak-measurement followed by post-selection can change the boundary condition at the temporal boundary in the path-integral representation of a density matrix, which will lead to different behaviors of the "strange correlator".
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Decades of research have revealed a deep understanding of topological quantum matter with protected edge modes. We report that even richer physics emerges when tuning between two topological phases of matter whose respective edge modes are incompatible. The frustration at the edge leads to novel boundary physics, such as symmetry-breaking phases with exotic non-Landau transitions---even when the edge is zero-dimensional. As a minimal case study, we consider spin chains with ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}$ symmetry, exhibiting two nontrivial symmetry-protected topological (SPT) phases. At the bulk 1+1D critical transition between these SPT phases, we find two stable 0+1D boundary phases, each spontaneously breaking one of the Z3{\mathbb{Z}}_{3}Z3​ symmetries. Furthermore, we find that a single boundary parameter tunes a non-Landau boundary critical transition between these two phases. This constitutes a 0+1D version of an exotic phenomenon driven by charged vortex condensation known as deconfined quantum criticality. This work highlights the rich unexplored physics of criticality between nontrivial topological phases and provides insights into the burgeoning field of gapless topological phases.
In symmetry protected topological (SPT) phases, the combination of symmetries and a bulk gap stabilizes protected modes at surfaces or at topological defects. Understanding the fate of these modes at a quantum critical point, when the protecting symmetries are on the verge of being broken, is an outstanding problem. This interplay of topology and criticality must incorporate both the bulk dynamics of critical points, often described by nontrivial conformal field theories, and SPT physics. Here, we study the simplest nontrivial setting - that of a 0+1 dimensional topological mode - a quantum spin - coupled to a 2+1D critical bulk. Using the large-NNN technique we solve a series of models which, as a consequence of topology, demonstrate intermediate coupling fixed points. We compare our results to previous numerical simulations and find good agreement. We also point out intriguing connections to generalized Kondo problems and Sachdev-Ye-Kitaev (SYK) models. In particular, we show that a Luttinger theorem derived for the complex SYK models, that relates the charge density to particle-hole asymmetry, also holds in our setting. These results should help stimulate further analytical study of the interplay between SPT physics and quantum criticality.
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