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arXiv:2201.07023 · Physical review. B./Physical review. B

Strongly correlated crystalline higher-order topological phases in two-dimensional systems: A coupled-wire study

Jian-Hao Zhang

Published Jul 22, 2022• 12 citations
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Abstract

Coupled-wire constructions have been widely applied to quantum Hall systems and symmetry-protected topological (SPT) phases. In this Letter, we use the coupled one-dimensional nonchiral Luttinger liquids with domain-wall structured mass terms as quantum wires to construct crystalline higher-order topological superconductors (HOTSCs) in two-dimensional (2D) interacting fermionic systems by two representative examples: a ${D}_{4}$-symmetric class-D HOTSC and a ${C}_{4}$-symmetric class-BDI HOTSC, with Majorana corner modes on the edge. Furthermore, based on the coupled-wire constructions, the quantum phase transitions between different phases of 2D HOTSCs by tuning the interwire coupling are investigated in a straightforward way.

DOI
10.1103/physrevb.106.l020503

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Coupled-wire constructions have been widely applied to quantum Hall systems and symmetry-protected topological (SPT) phases. In this Letter, we use the coupled one-dimensional nonchiral Luttinger liquids with domain-wall structured mass terms as quantum wires to construct crystalline higher-order topological superconductors (HOTSCs) in two-dimensional (2D) interacting fermionic systems by two representative examples: a D4{D}_{4}D4​-symmetric class-D HOTSC and a C4{C}_{4}C4​-symmetric class-BDI HOTSC, with Majorana corner modes on the edge. Furthermore, based on the coupled-wire constructions, the quantum phase transitions between different phases of 2D HOTSCs by tuning the interwire coupling are investigated in a straightforward way.
We propose a minimal interacting lattice model for two-dimensional class-DDD higher-order topological superconductors with no free-fermion counterpart. A Lieb-Schultz-Mattis-type constraint has been proposed and applied to guide our lattice model construction. Our model exhibits a trivial product ground state in the weakly interacting regime, whereas, increasing electron correlations provoke a novel topological quantum phase transition to a D4{D}_{4}D4​-symmetric higher-order topological superconducting state. The symmetry-protected Majorana corner modes are numerically confirmed with the matrix-product-state technique. Our theory paves the way for studying correlated higher-order topology with explicit lattice model constructions.
Crystalline symmetry and time-reversal symmetry are commonly present in real superconducting materials. However, the topological classification of systems respecting these symmetries, particularly for interacting fermions, remains incomplete. In this work, we systematically classify time-reversal symmetry-protected crystalline topological superconductors in two-dimensional interacting fermionic systems using an explicit real-space construction. Among the resulting phases, we identify intrinsically interacting fermionic topological superconductors, i.e., phases that cannot be realized in either free-fermion or interacting bosonic systems. For spinless fermions with protecting symmetry group C4×Z2TC_4 \times Z_2^TC4​×Z2T​ or D4×Z2TD_4 \times Z_2^TD4​×Z2T​ (plus fermion parity), the intrinsic sector has a Z4Z_4Z4​ classification. The corresponding root phases generating this Z4Z_4Z4​ classification admit a transparent real-space construction in terms of decorated 1D blocks. These blocks are 1D fermionic symmetry-protected topological (FSPT) phases, realizable as double Majorana chains. We further find the corresponding Z4Z_4Z4​ spinless intrinsic phases for wallpaper groups p4p4p4, p4mp4mp4m, and p4gp4gp4g. We also find an additional Z2Z_2Z2​ intrinsically interacting phase for spinless fermions with wallpaper group pmpmpm, which is absent with the corresponding point-group symmetry alone. Moreover, these intrinsic phases naturally give rise to higher-order FSPT phases that support corner zero modes. Finally, we verify the crystalline equivalence principle for generic 2D interacting FSPT systems with both crystalline and internal symmetries.
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.
This dissertation discusses some applications of conformal field theories (CFTs) in topological phases of matter. The first part is devoted to a discussion of coupled wire constructions of some novel quantum Hall systems. Through a theoretical coupled wire model, we construct strongly correlated electronic integer quantum Hall states with filling factor 16. The edge state is a bosonic chiral (E8)1 CFT, which is closely related to topological paramagnets in (3+1)d. As a distinguishing feature, these states support electric and thermal Hall transport violating the Wiedemann-Franz law. We further construct two descendant non-Abelian quantum Hall states at filling 8, each carrying bosonic chiral (G2)1 or (F4)1 edge theories, and hosting Fibonacci anyonic excitations in the bulk. Finally, we discover a new notion of particle-hole conjugation based on the E8 state that relates the G2 and F4 Fibonacci states, which is reminiscent of similar physics in half-filled Landau level. The second part is focused on the surface topological orders of 3D bulk topological systems. Symmetry protected and symmetry-enriched topological (SPT/SET) phases in three dimensions are quantum systems that support non-trivial two-dimensional surface states. These surface states develop finite excitation energy gaps when the relevant symmetries are broken. On the other hand, one-dimensional gapless modes can populate along interfaces that separate adjacent gapped surface domains with distinct symmetry-breaking orders. A surface strip pattern in general reduces the low-energy SPT/SET surface degrees of freedom onto a 2D array of gapless 1D channels. These channels can be coupled to one another by quasiparticle tunneling, and these inter-wire interactions collectively provide an effective description of the surface state. In this part, we study a general class of symmetry-preserving or breaking SPT/SET surface states that admit finite excitation energy gaps and Abelian topological orders via the coupled wire construction. In particular, we focus on the prototype Abelian surface topological orders that fall under the ADE classification of simply-laced Lie algebras. We also elaborate on the emergent symmetry and duality properties of the coupled wire models. The third part is to discuss the relation between the conformal boundary state and (2+1)d SPT phases. We propose a diagnostic tool for detecting nontrivial symmetry-protected topological (SPT) phases protected by a symmetry group G in 2 + 1 dimensions. Our method is based on directly studying the 1 + 1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests as the non-existence of boundary states in the CFT that preserves both the conformal symmetry and the global symmetry G. We discuss the relation between edgeability and gappability in the presence of G. We study several examples including time-reversal symmetric topological insulators, ZN symmetric bosonic SPT phases, and Z2 × Z2 symmetric topological superconductors.
The construction and classification of crystalline symmetry protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Crystalline SPT phases are not only of conceptual importance, but also provide us great opportunities towards experimental realization since space group symmetries naturally exist for any realistic material. In this paper, we systematically classify the crystalline topological superconductors (TSC) and topological insulators (TI) in 2D interacting fermionic systems by using an explicit real-space construction. In particular, we discover an intriguing fermionic crystalline topological superconductor that can only be realized in interacting fermionic systems (i.e., not in free-fermion or interacting bosonic systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for generic 2D interacting fermionic systems.
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 $(d\ensuremath{-}1)$-dimensional surface states, HOSPT phases in ddd 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.