Qing-Rui Wang
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
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Research Interests
Qing-Rui Wang has a total of 14 co-authors. Here are the top co-authors by number of shared publications:
- Zheng-Cheng Gu
Chinese University of Hong Kong
5 shared publications - Peng Ye
School of Physics Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices State Key Laboratory of Optoelectronic Materials and Technologies Sun Yat-sen University 510275 Guangzhou China
3 shared publications - Chenjie Wang
Department of Physics and HK Institute of Quantum Science & Technology, The University of Hong Kong 3 , Pokfulam Road, Hong Kong
2 shared publications - Jing-Ren Zhou
The Chinese University of Hong Kong
2 shared publications Meng Cheng
Department of Physics, Yale University, New Haven, Connecticut 06511-8499, USA
2 shared publicationsShang-Qiang Ning
Department of Physics The Hong Kong University of Science and Technology Clear Water Bay Kowloon Hong Kong China
2 shared publications
Latest Publications
- Open Access
Classification of Interacting Topological Crystalline Superconductors in Three Dimensions and Beyond
Although classification for free-fermion topological superconductors (TSC) is established, systematically understanding the classification of 3D interacting TSCs remains difficult, especially those protected by crystalline symmetries like the 230 space groups. We build up a general framework for systematically classifying 3D interacting TSCs protected by crystalline symmetries together with discrete internal symmetries. We first establish a complete classification for fermionic symmetry protected topological phases (FSPT) with purely discrete internal symmetries, which determines the crystalline case via the crystalline equivalence principle. Using domain wall decoration, we obtain classification data and formulas for generic FSPTs, what are suitable for systematic computation. The four layers of decoration data $(n_1, n_2, n_3, ν_4)$ characterize a 3D FSPT with symmetry $G_b\times_{ω_2}Z_2^f$, corresponding to $p+ip$, Kitaev chain, complex fermion, and bosonic SPT layers. Inspired by previous works, a crucial aspect is the $p+ip$ layer, where classification involves two possibilities: anti-unitary and infinite-order symmetries (e.g., translation). We show the former maps to some mirror FSPT classification with the mirror plane decorated by a $p+ip$ superconductor, while the latter is determined by the free part of $H^1(G_b, Z_T)$, corresponding to weak TSCs. Another key point is the Kitaev chain decoration for the anti-unitary symmetries, which differs essentially from unitary ones. We explicitly obtain formulas for all three layers of decoration $(n_2, n_3, ν_4)$, which are amenable to automatic computation. As an application, we classify the 230 space-group topological crystalline superconductors in interacting electronic systems.Open AccessClassification of Interacting Topological Crystalline Superconductors in Three Dimensions and Beyond
Although classification for free-fermion topological superconductors (TSC) is established, systematically understanding the classification of 3D interacting TSCs remains difficult, especially those protected by crystalline symmetries like the 230 space groups. We build up a general framework for systematically classifying 3D interacting TSCs protected by crystalline symmetries together with discrete internal symmetries. We first establish a complete classification for fermionic symmetry protected topological phases (FSPT) with purely discrete internal symmetries, which determines the crystalline case via the crystalline equivalence principle. Using domain wall decoration, we obtain classification data and formulas for generic FSPTs, what are suitable for systematic computation. The four layers of decoration data $(n_1, n_2, n_3, ν_4)$ characterize a 3D FSPT with symmetry $G_b\times_{ω_2}Z_2^f$, corresponding to $p+ip$, Kitaev chain, complex fermion, and bosonic SPT layers. Inspired by previous works, a crucial aspect is the $p+ip$ layer, where classification involves two possibilities: anti-unitary and infinite-order symmetries (e.g., translation). We show the former maps to some mirror FSPT classification with the mirror plane decorated by a $p+ip$ superconductor, while the latter is determined by the free part of $H^1(G_b, Z_T)$, corresponding to weak TSCs. Another key point is the Kitaev chain decoration for the anti-unitary symmetries, which differs essentially from unitary ones. We explicitly obtain formulas for all three layers of decoration $(n_2, n_3, ν_4)$, which are amenable to automatic computation. As an application, we classify the 230 space-group topological crystalline superconductors in interacting electronic systems. - 2 May 2025
- Published by American Physical Society (APS) in Physical Review B
- Vol. 111 (20)
Systematic construction of interfaces and anomalous boundaries for fermionic symmetry-protected topological phases
- 2 May 2025
- Published by American Physical Society (APS) in Physical Review B
- Vol. 111 (20)
Systematic construction of interfaces and anomalous boundaries for fermionic symmetry-protected topological phases
- Open Access
- 3 November 2023
- Published by American Physical Society (APS) in Physical Review Research
- Vol. 5 (4) , 043111
Continuum field theory of three-dimensional topological orders with emergent fermions and braiding statistics
Universal topological data of topologically ordered phases can be captured by topological quantum field theory in continuous space time by taking the limit of low energies and long wavelengths. While previous continuum field-theoretical studies of topological orders in three-dimensional (3D) real space focused on either self-statistics, braiding statistics, shrinking rules, fusion rules, or quantum dimensions, it has yet to systematically put all topological data together in a unified continuum field-theoretical framework. Here, we construct the topological field theory with twisted terms (e.g., and ) as well as a -matrix term to simultaneously explore all such topological data and reach anomaly-free topological orders. Following the spirit of the famous -matrix Chern-Simons theory of two-dimensional topological orders, we present general formulas and systematically show how the -matrix term confines topological excitations, and how self-statistics of particles is transmuted between bosonic one and fermionic one. To reach anomaly-free topological orders, we explore, within the present continuum field-theoretical framework, how the principle of gauge invariance fundamentally influences possible realizations of topological data. More concretely, we present the topological actions of (i) particle-loop braidings with emergent fermions, (ii) multiloop braidings with emergent fermions, and (iii) Borromean rings braidings with emergent fermions, and calculate their universal topological data. Together with previous efforts, our paper paves the way toward a more systematic and complete continuum field-theoretical analysis of exotic topological properties of 3D topological orders. Several interesting future directions are also discussed.- 3 November 2023
- Published by American Physical Society (APS) in Physical Review Research
- Vol. 5 (4) , 043111
Open AccessContinuum field theory of three-dimensional topological orders with emergent fermions and braiding statistics
Universal topological data of topologically ordered phases can be captured by topological quantum field theory in continuous space time by taking the limit of low energies and long wavelengths. While previous continuum field-theoretical studies of topological orders in three-dimensional (3D) real space focused on either self-statistics, braiding statistics, shrinking rules, fusion rules, or quantum dimensions, it has yet to systematically put all topological data together in a unified continuum field-theoretical framework. Here, we construct the topological field theory with twisted terms (e.g., and ) as well as a -matrix term to simultaneously explore all such topological data and reach anomaly-free topological orders. Following the spirit of the famous -matrix Chern-Simons theory of two-dimensional topological orders, we present general formulas and systematically show how the -matrix term confines topological excitations, and how self-statistics of particles is transmuted between bosonic one and fermionic one. To reach anomaly-free topological orders, we explore, within the present continuum field-theoretical framework, how the principle of gauge invariance fundamentally influences possible realizations of topological data. More concretely, we present the topological actions of (i) particle-loop braidings with emergent fermions, (ii) multiloop braidings with emergent fermions, and (iii) Borromean rings braidings with emergent fermions, and calculate their universal topological data. Together with previous efforts, our paper paves the way toward a more systematic and complete continuum field-theoretical analysis of exotic topological properties of 3D topological orders. Several interesting future directions are also discussed. - 6 September 2023
- Published by American Physical Society (APS) in Physical Review B
- Vol. 108 (12)
Exactly solvable lattice models for interacting electronic insulators in two dimensions
- 6 September 2023
- Published by American Physical Society (APS) in Physical Review B
- Vol. 108 (12)
Exactly solvable lattice models for interacting electronic insulators in two dimensions
- Open Access
Continuum field theory of 3D topological orders with emergent fermions and braiding statistics
Universal topological data of topologically ordered phases can be captured by topological quantum field theory in continuous space time by taking the limit of low energies and long wavelengths. While previous continuum field-theoretical studies of topological orders in $3$D real space focus on either self-statistics, braiding statistics, shrinking rules, fusion rules or quantum dimensions, it is yet to systematically put all topological data together in a unified continuum field-theoretical framework. Here, we construct the topological $BF$ field theory with twisted terms (e.g., $AAdA$ and $AAB$) as well as a $K$-matrix $BB$ term, in order to simultaneously explore all such topological data and reach anomaly-free topological orders. Following the spirit of the famous $K$-matrix Chern-Simons theory of $2$D topological orders, we present general formulas and systematically show how the $K$-matrix $BB$ term confines topological excitations, and how self-statistics of particles is transmuted between bosonic one and fermionic one. In order to reach anomaly-free topological orders, we explore, within the present continuum field-theoretical framework, how the principle of gauge invariance fundamentally influences possible realizations of topological data. More concretely, we present the topological actions of (i) particle-loop braidings with emergent fermions, (ii) multiloop braidings with emergent fermions, and (iii) Borromean-Rings braidings with emergent fermions, and calculate their universal topological data. Together with the previous efforts, our work paves the way toward a more systematic and complete continuum field-theoretical analysis of exotic topological properties of $3$D topological orders. Several interesting future directions are also discussed.Open AccessContinuum field theory of 3D topological orders with emergent fermions and braiding statistics
Universal topological data of topologically ordered phases can be captured by topological quantum field theory in continuous space time by taking the limit of low energies and long wavelengths. While previous continuum field-theoretical studies of topological orders in $3$D real space focus on either self-statistics, braiding statistics, shrinking rules, fusion rules or quantum dimensions, it is yet to systematically put all topological data together in a unified continuum field-theoretical framework. Here, we construct the topological $BF$ field theory with twisted terms (e.g., $AAdA$ and $AAB$) as well as a $K$-matrix $BB$ term, in order to simultaneously explore all such topological data and reach anomaly-free topological orders. Following the spirit of the famous $K$-matrix Chern-Simons theory of $2$D topological orders, we present general formulas and systematically show how the $K$-matrix $BB$ term confines topological excitations, and how self-statistics of particles is transmuted between bosonic one and fermionic one. In order to reach anomaly-free topological orders, we explore, within the present continuum field-theoretical framework, how the principle of gauge invariance fundamentally influences possible realizations of topological data. More concretely, we present the topological actions of (i) particle-loop braidings with emergent fermions, (ii) multiloop braidings with emergent fermions, and (iii) Borromean-Rings braidings with emergent fermions, and calculate their universal topological data. Together with the previous efforts, our work paves the way toward a more systematic and complete continuum field-theoretical analysis of exotic topological properties of $3$D topological orders. Several interesting future directions are also discussed. - 10 April 2023
- Published by American Physical Society (APS) in Physical Review B
- Vol. 107 (16) , 165117
Non-Abelian fusion, shrinking, and quantum dimensions of Abelian gauge fluxes
Braiding and fusion rules of topological excitations are indispensable topological invariants in topological quantum computation and topological orders. While excitations in two dimensions (2D) are always particlelike anyons, those in three dimensions (3D) incorporate not only particles but also loops, spatially nonlocal objects, making it novel and challenging to study topological invariants in higher dimensions. While 2D fusion rules have been well understood from bulk Chern-Simons field theory and edge conformal field theory, it is yet to be thoroughly explored for 3D fusion rules from higher-dimensional bulk topological field theory. Here, we perform a field-theoretical study on (i) how loops that carry Abelian gauge fluxes fuse and (ii) how loops are shrunk into particles in the path integral, which generates fusion rules, loop-shrinking rules, and descendent invariants, e.g., quantum dimensions. We first assign a gauge-invariant Wilson operator to each excitation and determine the number of distinct excitations through equivalence classes of Wilson operators. Then, we adiabatically shift two Wilson operators together to observe how they fuse and are split in the path integral; despite the Abelian nature of the gauge fluxes carried by loops, their fusions may be of non-Abelian nature. Meanwhile, we adiabatically deform world sheets of unknotted loops into world lines and examine the shrinking outcomes; we find that the resulting loop-shrinking rules are algebraically consistent to fusion rules. Interestingly, fusing a pair of loop and antiloop may generate multiple vacua, but fusing a pair of anyon and antianyon in 2D has one vacuum only. By establishing a field-theoretical ground for fusion and shrinking in 3D, this work leaves intriguing directions for future exploration, e.g., symmetry enrichment, quantum gates, and topological order of braided monoidal 2-category of 2-group.
- 10 April 2023
- Published by American Physical Society (APS) in Physical Review B
- Vol. 107 (16) , 165117
Non-Abelian fusion, shrinking, and quantum dimensions of Abelian gauge fluxes
Braiding and fusion rules of topological excitations are indispensable topological invariants in topological quantum computation and topological orders. While excitations in two dimensions (2D) are always particlelike anyons, those in three dimensions (3D) incorporate not only particles but also loops, spatially nonlocal objects, making it novel and challenging to study topological invariants in higher dimensions. While 2D fusion rules have been well understood from bulk Chern-Simons field theory and edge conformal field theory, it is yet to be thoroughly explored for 3D fusion rules from higher-dimensional bulk topological field theory. Here, we perform a field-theoretical study on (i) how loops that carry Abelian gauge fluxes fuse and (ii) how loops are shrunk into particles in the path integral, which generates fusion rules, loop-shrinking rules, and descendent invariants, e.g., quantum dimensions. We first assign a gauge-invariant Wilson operator to each excitation and determine the number of distinct excitations through equivalence classes of Wilson operators. Then, we adiabatically shift two Wilson operators together to observe how they fuse and are split in the path integral; despite the Abelian nature of the gauge fluxes carried by loops, their fusions may be of non-Abelian nature. Meanwhile, we adiabatically deform world sheets of unknotted loops into world lines and examine the shrinking outcomes; we find that the resulting loop-shrinking rules are algebraically consistent to fusion rules. Interestingly, fusing a pair of loop and antiloop may generate multiple vacua, but fusing a pair of anyon and antianyon in 2D has one vacuum only. By establishing a field-theoretical ground for fusion and shrinking in 3D, this work leaves intriguing directions for future exploration, e.g., symmetry enrichment, quantum gates, and topological order of braided monoidal 2-category of 2-group.
- 13 December 2022
- Published by American Physical Society (APS) in Physical Review B
- Vol. 106 (24) , 245120
Towards a complete classification of nonchiral topological phases in two-dimensional fermion systems
In recent years, fermionic topological phases of quantum matter has attracted a lot of attention. In a pioneer work by Gu, Wang, and Wen, the concept of equivalence classes of fermionic local unitary (FLU) transformations was proposed to systematically understand nonchiral topological phases in 2D fermion systems and an incomplete classification was obtained. On the other hand, the physical picture of fermion condensation and its corresponding super pivotal categories give rise to a generic mathematical framework to describe fermionic topological phases of quantum matter. In particular, it has been pointed out that in certain fermionic topological phases, there exists the so-called q-type anyon excitations, which have no analogues in bosonic theories. In this paper, we generalize the Gu, Wang, and Wen construction to include those fermionic topological phases with q-type anyon excitations. We argue that all nonchiral fermionic topological phases in are characterized by a set of tensors , which satisfy a set of nonlinear algebraic equations parameterized by phase factors and . Moreover, consistency conditions among algebraic equations give rise to additional constraints on these phase factors, which allow us to construct a topological invariant partition for an arbitrary triangulation of 3D spin manifold. Finally, several examples with q-type anyon excitations are discussed, including the fermionic topological phase from Tambara-Yamagami category for , which can be regarded as the parafermion generalization of Ising fermionic topological phase.- 13 December 2022
- Published by American Physical Society (APS) in Physical Review B
- Vol. 106 (24) , 245120
Towards a complete classification of nonchiral topological phases in two-dimensional fermion systems
In recent years, fermionic topological phases of quantum matter has attracted a lot of attention. In a pioneer work by Gu, Wang, and Wen, the concept of equivalence classes of fermionic local unitary (FLU) transformations was proposed to systematically understand nonchiral topological phases in 2D fermion systems and an incomplete classification was obtained. On the other hand, the physical picture of fermion condensation and its corresponding super pivotal categories give rise to a generic mathematical framework to describe fermionic topological phases of quantum matter. In particular, it has been pointed out that in certain fermionic topological phases, there exists the so-called q-type anyon excitations, which have no analogues in bosonic theories. In this paper, we generalize the Gu, Wang, and Wen construction to include those fermionic topological phases with q-type anyon excitations. We argue that all nonchiral fermionic topological phases in are characterized by a set of tensors , which satisfy a set of nonlinear algebraic equations parameterized by phase factors and . Moreover, consistency conditions among algebraic equations give rise to additional constraints on these phase factors, which allow us to construct a topological invariant partition for an arbitrary triangulation of 3D spin manifold. Finally, several examples with q-type anyon excitations are discussed, including the fermionic topological phase from Tambara-Yamagami category for , which can be regarded as the parafermion generalization of Ising fermionic topological phase.- Open Access
Non-Abelian Fusion, Shrinking and Quantum Dimensions of Abelian Gauge Fluxes
Braiding and fusion rules of topological excitations are indispensable topological invariants in topological quantum computation and topological orders. While excitations in 2D are always particle-like anyons, those in 3D incorporate not only particles but also loops -- spatially nonlocal objects -- making it novel and challenging to study topological invariants in higher dimensions. While 2D fusion rules have been well understood from bulk Chern-Simons field theory and edge conformal field theory, it is yet to be thoroughly explored for 3D fusion rules from higher dimensional bulk topological field theory. Here, we perform a field-theoretical study on (i) how loops that carry Abelian gauge fluxes fuse and (ii) how loops are shrunk into particles in the path integral, which generates fusion rules, loop-shrinking rules, and descendent invariants, e.g., quantum dimensions. We first assign a gauge-invariant Wilson operator to each excitation and determine the number of distinct excitations through equivalence classes of Wilson operators. Then, we adiabatically shift two Wilson operators together to observe how they fuse and are split in the path integral; despite the Abelian nature of the gauge fluxes carried by loops, their fusions may be of non-Abelian nature. Meanwhile, we adiabatically deform world-sheets of unknotted loops into world-lines and examine the shrinking outcomes; we find that the resulting loop-shrinking rules are algebraically consistent to fusion rules. Interestingly, fusing a pair of loop and anti-loop may generate multiple vacua, but fusing a pair of anyon and anti-anyon in 2D has one vacuum only. By establishing a field-theoretical ground for fusion and shrinking in 3D, this work leaves intriguing directions, e.g., symmetry enrichment, quantum gates, and physics of braided monoidal 2-category of 2-group.
Open AccessNon-Abelian Fusion, Shrinking and Quantum Dimensions of Abelian Gauge Fluxes
Braiding and fusion rules of topological excitations are indispensable topological invariants in topological quantum computation and topological orders. While excitations in 2D are always particle-like anyons, those in 3D incorporate not only particles but also loops -- spatially nonlocal objects -- making it novel and challenging to study topological invariants in higher dimensions. While 2D fusion rules have been well understood from bulk Chern-Simons field theory and edge conformal field theory, it is yet to be thoroughly explored for 3D fusion rules from higher dimensional bulk topological field theory. Here, we perform a field-theoretical study on (i) how loops that carry Abelian gauge fluxes fuse and (ii) how loops are shrunk into particles in the path integral, which generates fusion rules, loop-shrinking rules, and descendent invariants, e.g., quantum dimensions. We first assign a gauge-invariant Wilson operator to each excitation and determine the number of distinct excitations through equivalence classes of Wilson operators. Then, we adiabatically shift two Wilson operators together to observe how they fuse and are split in the path integral; despite the Abelian nature of the gauge fluxes carried by loops, their fusions may be of non-Abelian nature. Meanwhile, we adiabatically deform world-sheets of unknotted loops into world-lines and examine the shrinking outcomes; we find that the resulting loop-shrinking rules are algebraically consistent to fusion rules. Interestingly, fusing a pair of loop and anti-loop may generate multiple vacua, but fusing a pair of anyon and anti-anyon in 2D has one vacuum only. By establishing a field-theoretical ground for fusion and shrinking in 3D, this work leaves intriguing directions, e.g., symmetry enrichment, quantum gates, and physics of braided monoidal 2-category of 2-group.
- 6 September 2022
- Published by American Physical Society (APS) in Physical Review B
- Vol. 106 (11) , 115104
Exactly solvable models for U(1) symmetry-enriched topological phases
We propose a general construction of commuting projector lattice models for 2D and 3D topological phases enriched by U(1) symmetry, with finite-dimensional Hilbert space per site. The construction starts from a commuting projector model of the topological phase and decorates U(1) charges to the state space in a consistent manner. We show that all 2D U(1) symmetry-enriched topological phases, which allow gapped boundaries without breaking the symmetry, can be realized through our construction. We also construct a large class of 3D topological phases with U(1) symmetry fractionalized on particles or loop excitations.
- 6 September 2022
- Published by American Physical Society (APS) in Physical Review B
- Vol. 106 (11) , 115104
Exactly solvable models for U(1) symmetry-enriched topological phases
We propose a general construction of commuting projector lattice models for 2D and 3D topological phases enriched by U(1) symmetry, with finite-dimensional Hilbert space per site. The construction starts from a commuting projector model of the topological phase and decorates U(1) charges to the state space in a consistent manner. We show that all 2D U(1) symmetry-enriched topological phases, which allow gapped boundaries without breaking the symmetry, can be realized through our construction. We also construct a large class of 3D topological phases with U(1) symmetry fractionalized on particles or loop excitations.
- 26 August 2021
- Published by American Physical Society (APS) in Physical Review B
- Vol. 104 (7) , 075151
Edge theories of two-dimensional fermionic symmetry protected topological phases protected by unitary Abelian symmetries
Abelian Chern-Simons theory, characterized by the so-called matrix, has been quite successful in characterizing and classifying Abelian fractional quantum Hall effect as well as symmetry protected topological (SPT) phases, especially for bosonic SPT phases. However, there are still some puzzles in dealing with fermionic SPT (fSPT) phases. In this paper, we utilize the Abelian Chern-Simons theory to study the fSPT phases protected by arbitrary Abelian total symmetry . Comparing to the bosonic SPT phases, fSPT phases with Abelian total symmetry have three new features: (1) they may support gapless Majorana fermion edge modes, (2) some nontrivial bosonic SPT phases may be trivialized if is a nontrivial extension of bosonic symmetry by , and (3) certain intrinsic fSPT phases can only be realized in interacting fermionic system. We obtain edge theories for various fSPT phases, which can also be regarded as conformal field theories with proper symmetry anomaly. In particular, we discover the construction of Luttinger liquid edge theories with central charge for type-III bosonic SPT phases protected by symmetry and the Luttinger liquid edge theories for intrinsically interacting fSPT protected by unitary Abelian symmetry. The ideas and methods used in these examples could be generalized to derive the edge theories of fSPT phases with arbitrary unitary Abelian total symmetry .- 26 August 2021
- Published by American Physical Society (APS) in Physical Review B
- Vol. 104 (7) , 075151
Edge theories of two-dimensional fermionic symmetry protected topological phases protected by unitary Abelian symmetries
Abelian Chern-Simons theory, characterized by the so-called matrix, has been quite successful in characterizing and classifying Abelian fractional quantum Hall effect as well as symmetry protected topological (SPT) phases, especially for bosonic SPT phases. However, there are still some puzzles in dealing with fermionic SPT (fSPT) phases. In this paper, we utilize the Abelian Chern-Simons theory to study the fSPT phases protected by arbitrary Abelian total symmetry . Comparing to the bosonic SPT phases, fSPT phases with Abelian total symmetry have three new features: (1) they may support gapless Majorana fermion edge modes, (2) some nontrivial bosonic SPT phases may be trivialized if is a nontrivial extension of bosonic symmetry by , and (3) certain intrinsic fSPT phases can only be realized in interacting fermionic system. We obtain edge theories for various fSPT phases, which can also be regarded as conformal field theories with proper symmetry anomaly. In particular, we discover the construction of Luttinger liquid edge theories with central charge for type-III bosonic SPT phases protected by symmetry and the Luttinger liquid edge theories for intrinsically interacting fSPT protected by unitary Abelian symmetry. The ideas and methods used in these examples could be generalized to derive the edge theories of fSPT phases with arbitrary unitary Abelian total symmetry .