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arXiv:1809.07325 · Physical review. B./Physical review. B

Classification and construction of higher-order symmetry-protected topological phases of interacting bosons

Alex Rasmussen, Yuan-Ming Lu

Published Feb 25, 2020• 65 citations
arXivHTMLPDF

Abstract

Motivated by the recent discovery of higher-order topological insulators, we study their counterparts in strongly interacting bosons: ``higher-order symmetry-protected topological (HOSPT) phases.'' While the usual (first-order) SPT phases in $d$ spatial dimensions support anomalous $(d\ensuremath{-}1)$-dimensional surface states, HOSPT phases in $d$ dimensions are characterized by topological boundary states of dimension $(d\ensuremath{-}2)$ or smaller, protected by certain global symmetries and robust against disorders. Based on a dimensional reduction analysis, we show that HOSPT phases can be built from lower-dimensional SPT phases in a way that preserves the associated crystalline symmetries. When the total symmetry is a direct product of global and crystalline symmetry groups, we are able to classify the HOSPT phases using the K\"unneth formula of group cohomology. Based on a decorated domain-wall picture of the K\"unneth formula, we show how to systematically construct the HOSPT phases, and demonstrate our construction with many examples in two and three dimensions.

DOI
10.1103/physrevb.101.085137

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Motivated by the recent discovery of higher-order topological insulators, we study their counterparts in strongly interacting bosons: ``higher-order symmetry-protected topological (HOSPT) phases.'' While the usual (first-order) SPT phases in ddd spatial dimensions support anomalous (d\ensuremath−1)(d\ensuremath{-}1)(d\ensuremath−1)-dimensional surface states, HOSPT phases in ddd dimensions are characterized by topological boundary states of dimension (d\ensuremath−2)(d\ensuremath{-}2)(d\ensuremath−2) or smaller, protected by certain global symmetries and robust against disorders. Based on a dimensional reduction analysis, we show that HOSPT phases can be built from lower-dimensional SPT phases in a way that preserves the associated crystalline symmetries. When the total symmetry is a direct product of global and crystalline symmetry groups, we are able to classify the HOSPT phases using the K"unneth formula of group cohomology. Based on a decorated domain-wall picture of the K"unneth formula, we show how to systematically construct the HOSPT phases, and demonstrate our construction with many examples in two and three dimensions.
The construction and classification of symmetry-protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Very recently, a complete classification and construction of space-group SPT phases were also proposed for interacting bosonic systems. In this Rapid Communication, we attempt to generalize this classification and construction scheme systematically into interacting fermion systems. In particular, we construct and classify point-group SPT phases for two-dimensional (2D) interacting fermion systems via lower-dimensional block-state decorations. We discover several intriguing fermionic SPT states that can only be realized in interacting fermion systems (i.e., not in free-fermion or bosonic SPT systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for 2D interacting fermion systems. Finally, the potential experimental realization of these different classes of point-group SPT phases in 2D correlated superconductors is addressed.
A wide variety of higher-order symmetry protected topological phase(HOSPT) with gapless corners or hinges had been proposed as a descendant of topological crystalline insulator protected by spatial symmetry. In this work, we address a new class of higher-order topological state which does not require crystalline symmetries but instead relies on subsystem symmetry for protection. We propose several strong interacting models with gapless hinge or corner based on a `decorated hinge-wall condensate' picture. The hinge-wall, which appears as the defect configuration of Z2Z_2Z2​ paramagnet is decorated with lower-dim SPT state. Such unique hinge-wall decoration structure leads to gapped surfaces separated by gapless hinges. The non-trivial nature of the hinge modes can be captured by a 1+11+11+1D conformal field theory with a Wess-Zumino-Witten term. Besides, we establish a no-go theorem to demonstrate the ungappable nature of the hinge by making a connection between generalized Lieb-Schultz-Mattis theorem and the boundary anomaly of HOSPT state. This universal correspondence engenders a comprehensive criterion to determine the existence of HOSPT under certain symmetry regardless of the microscopic Hamiltonian.
Abstract A wide variety of higher-order symmetry-protected topological phases (HOSPT) with gapless corners or hinges have been proposed as descendants of topological crystalline insulators protected by spatial symmetry. In this work, we address a new class of higher-order topological states that do not require crystalline symmetries but instead rely on subsystem symmetry for protection. We propose several strongly interacting models with gapless hinges or corners based on a decorated hinge-wall condensate picture. The hinge-wall, which appears as the defect configuration of a Z 2 paramagnet, is decorated with a lower-dimensional SPT state. Such a unique hinge-wall decoration structure leads to gapped surfaces separated by gapless hinges. The non-trivial nature of the hinge modes can be captured by a 1 + 1 D conformal field theory with a Wess–Zumino–Witten term. Moreover, we establish a no-go theorem to demonstrate the ungappable nature of the hinges by making a connection between the generalized Lieb–Schultz–Mattis theorem and the boundary anomaly of the HOSPT state. This universal correspondence engenders a comprehensive criterion to determine the existence of HOSPT under certain symmetries, regardless of the microscopic Hamiltonian.
It is demonstrated by explicit construction that three-dimensional bosonic crystalline symmetry protected topological (cSPT) phases are classified by ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})\ensuremath{\bigoplus}{H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ for all 230 space groups GGG, where ${H}_{\ensuremath{\phi}}^{n}(G;\mathbb{Z})$ denotes the nthn\mathrm{th}nth twisted group cohomology of GGG with Z\mathbb{Z}Z coefficients, and $\ensuremath{\phi}$ indicates that $g\ensuremath{\in}G$ acts nontrivially on coefficients by sending them to their inverses if ggg reverses space-time orientation and acts trivially otherwise. The previously known summand ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})$ corresponds only to crystalline phases built without the E8{E}_{8}E8​ state or its multiples on 2-cells of space. It is the crystalline analog of the ``group cohomology proposal'' for classifying bosonic symmetry protected topological (SPT) phases, which takes the form ${H}_{\ensuremath{\phi}}^{d+2}(G;\mathbb{Z})\ensuremath{\cong}{H}_{\ensuremath{\phi}}^{d+1}(G;\text{U}(1))$ for finite internal symmetry groups in ddd spatial dimensions. The new summand ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ classifies possible configurations of E8{E}_{8}E8​ states on 2-cells that can be used to build crystalline phases beyond the group cohomology proposal. The completeness of our classification and the physical meaning of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ are established through a combination of dimensional reduction, surface topological order, and explicit cellular construction. The value of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ can be easily read off from the international symbol for GGG. Our classification agrees with the prediction of the ``generalized cohomology hypothesis,'' which concerns the general structure of the classification of SPT phases, and therefore provides strong evidence for the validity of the said hypothesis in the realm of crystalline symmetries.
In this thesis, we study gapped topological phases of matter in systems with strong inter-particle interaction. They are challenging to analyze theoretically, because interaction not only gives rise to a plethora of phases that are otherwise absent, but also renders methods used to analyze non-interacting systems inadequate. By now, people have had a relatively systematic understanding of topological orders in two spatial dimensions. However, less is known about the higher dimensional cases. In Chapter 2, we will explore three dimensional long-range entangled topological orders in the framework of Walker-Wang models, which are a class of exactly solvable models for three-dimensional topological phases that are not known previously to be able to capture these phases. We find that they can represent a class of twisted discrete gauge theories, which were discovered using a different formalism. Meanwhile, a systematic theory of bosonic symmetry protected topological (SPT) phases in all spatial dimensions have been developed based on group cohomology. A generalization of the theory to group supercohomology has been proposed to classify and characterize fermionic SPT phases in all dimensions. However, it can only handle cases where the symmetry group of the system is a product of discrete unitary symmetries. Furthermore, the classification is known to be incomplete for certain symmetries. In Chapter 3, we will construct an exactly solvable model for the two-dimensional time-reversal-invariant topological superconductors, which could be valuable as a first attempt to a systematic understanding of strongly interacting fermionic SPT phases with anti-unitary symmetries in terms of exactly solvable models. In Chapter 4, we will propose an alternative classification of fermionic SPT phases using the spin cobordism theory, which hopefully can capture all the phases missing in the supercohomology classification. We test this proposal in the case of fermionic SPT phases with Z2 symmetry, where Z2 is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known fermionic SPT phases in space dimensions less than or equal to 3.
We use higher-dimensional bosonization and fermion decoration to construct exactly soluble interacting fermion models to realize fermionic symmetry-protected trivial (SPT) orders (which are also known as symmetry-protected topological orders) in any dimensions and for generic fermion symmetries Gf{G}_{f}Gf​, which can be a nontrivial Z2f{Z}_{2}^{f}Z2f​ extension ${Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}$ (where Z2f{Z}_{2}^{f}Z2f​ is the fermion-number-parity symmetry and Gb{G}_{b}Gb​ is the bosonic symmetry). This generalizes the previous results from group supercohomology of Gu and Wen (arXiv:1201.2648), where Gf{G}_{f}Gf​ is assumed to be ${Z}_{2}^{f}\ifmmode\times\else\texttimes\fi{}{G}_{b}$. We find that the (d+1)(d+1)(d+1)-dimensional [(d+1)D][(d+1)\mathrm{D}][(d+1)D] fermionic SPT phases with bosonic symmetry Gb{G}_{b}Gb​ and from fermion decoration construction can be described in a compact way using higher group homomorphism: $\mathcal{B}{G}_{b}\stackrel{\ensuremath{\varphi}}{\ensuremath{\rightarrow}}\mathcal{B}({Z}_{2},2;{Z}_{2},d)$. In fact, the fermion symmetry is more precisely described by the structure ${Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}S{O}_{\ensuremath{\infty}}$ (or ${Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}{O}_{\ensuremath{\infty}}$ with time-reversal symmetry). In this case the (d+1)D(d+1)\mathrm{D}(d+1)D fermionic SPT phases are better described by $\mathcal{B}({Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}S{O}_{\ensuremath{\infty}})\stackrel{\ensuremath{\varphi}}{\ensuremath{\rightarrow}}\mathcal{B}(S{O}_{\ensuremath{\infty}},1;{Z}_{2},d)$ [or $\mathcal{B}({Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}{O}_{\ensuremath{\infty}})\stackrel{\ensuremath{\varphi}}{\ensuremath{\rightarrow}}\mathcal{B}({O}_{\ensuremath{\infty}},1;{Z}_{2},d)]$.