Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2309.12389 · Physical review. B./Physical review. B

Characterization and classification of interacting ( 2+1 )-dimensional topological crystalline insulators with orientation-preserving wallpaper groups

Naren Manjunath, Vladimir Calvera, Maissam Barkeshli

Published Jan 31, 2024• 16 citations
arXivHTMLPDF

Abstract

While free fermion topological crystalline insulators have been largely classified, the analogous problem in the strongly interacting case has been only partially solved, and the relationship between the free and interacting classifications is not well understood. In this paper, we develop a characterization and classification of interacting, invertible fermionic topological phases in ($2+1$) dimensions with charge conservation, discrete magnetic translation, and $M$-fold point group rotation symmetries, which form the group ${G}_{f}=\text{U}{(1)}^{f}{\ifmmode\times\else\texttimes\fi{}}_{\ensuremath{\phi}}[{\mathbb{Z}}^{2}\ensuremath{\rtimes}{\mathbb{Z}}_{M}]$ for $M=1,2,3,4,\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}6$. $\ensuremath{\phi}$ is the magnetic flux per unit cell. We derive a topological response theory in terms of background crystalline gauge fields, which gives a complete classification of different phases and a physical characterization in terms of quantized response to symmetry defects. We then derive the same classification in terms of a set of real-space invariants ${{\mathrm{\ensuremath{\Theta}}}_{\text{o}}^{\ifmmode\pm\else\textpm\fi{}}}$ that can be obtained from ground state expectation values of suitable partial rotation operators. We explicitly relate these real-space invariants to the quantized coefficients in the topological response theory, and find the dependence of the invariants on the chiral central charge ${c}_{\ensuremath{-}}$ of the invertible phase. Finally, when $\ensuremath{\phi}=0,$ we derive an explicit map between the free and interacting classifications.

DOI
10.1103/physrevb.109.035168

Related papers

Interacting crystalline topological insulators in two-dimensions with time-reversal symmetry

Martina O. Soldini, Ömer M. Aksoy, Titus Neupert

Topology is routinely used to understand the physics of electronic insulators. However, for strongly interacting electronic matter, such as Mott insulators, a comprehensive topological characterization is still lacking. When their ground state only contains short-range entanglement and does not break symmetries spontaneously, they generically realize crystalline fermionic symmetry-protected topological phases (cFSPTs), supporting gapless modes at the boundaries or at the lattice defects. Here, we provide an exhaustive classification of cFSPTs in two dimensions with U ( 1 ) charge-conservation and spinful time-reversal symmetries, namely, those generically present in spin-orbit coupled insulators, for any of the 17 wallpaper groups. It has been shown that the classification of cFSPTs can be understood from appropriate real-space decorations of lower-dimensional subspaces, and we expose how these relate to the Wyckoff positions of the lattice. We find that all nontrivial one-dimensional decorations require electronic interactions. Furthermore, we provide model Hamiltonians for various decorations, and discuss the signatures of cFSPTs. This classification paves the way to further explore topological interacting insulators, providing the backbone information in generic model systems and ultimately in experiments. Published by the American Physical Society 2024

arXiv:2404.11650•Physical Review Research•Aug 22, 2024•3 citations

Interacting crystalline topological insulators in two-dimensions with time-reversal symmetry

Martina O. Soldini, Ömer M. Aksoy, Titus Neupert

Topology is routinely used to understand the physics of electronic insulators. However, for strongly interacting electronic matter, such as Mott insulators, a comprehensive topological characterization is still lacking. When their ground state only contains short-range entanglement and does not break symmetries spontaneously, they generically realize crystalline fermionic symmetry-protected topological phases (cFSPTs), supporting gapless modes at the boundaries or at the lattice defects. Here, we provide an exhaustive classification of cFSPTs in two dimensions with charge-conservation and spinful time-reversal symmetries, namely, those generically present in spin-orbit coupled insulators, for any of the 17 wallpaper groups. It has been shown that the classification of cFSPTs can be understood from appropriate real-space decorations of lower-dimensional subspaces, and we expose how these relate to the Wyckoff positions of the lattice. We find that all nontrivial one-dimensional decorations require electronic interactions. Furthermore, we provide model Hamiltonians for various decorations, and discuss the signatures of cFSPTs. This classification paves the way to further explore topological interacting insulators, providing the backbone information in generic model systems and ultimately in experiments.

•Aug 22, 2024•2 citations

Classification of Interacting Topological Crystalline Superconductors in Three Dimensions and Beyond

Shang-Qiang Ning, Xing-Yu Ren, Qing-Rui Wang +2 more

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.

arXiv:2512.25069•Dec 31, 2025•0 citations

Crystalline topological invariants in quantum many-body systems

Naren Manjunath, Maissam Barkeshli

Crystalline symmetries give rise to topological invariants that can distinguish quantum phases of matter. Understanding these in strongly interacting systems is an ongoing research direction requiring non-perturbative methods. Recent developments have demonstrated that even classic models, like the Harper-Hofstadter model of free fermions on a lattice in a magnetic field, yield a host of crystalline symmetry protected topological invariants. Here we review some of these developments, focusing mainly on how to characterize, classify, and detect invariants arising from lattice translation and rotation symmetries along with charge conservation in two-dimensional systems, including integer and fractional Chern insulators.

cond-mat.str-elCross-listedarXiv:2604.10338•Apr 14, 2026•0 citations

Interaction effects on the classification of crystalline topological insulators and superconductors

Xue-Yang Song, Andreas P. Schnyder

We classify interacting topological insulators and superconductors with order-two crystal symmetries (reflection or twofold rotation), focusing on the case where interactions reduce the noninteracting classification. We find that the free-fermion ${\mathbb{Z}}_{2}$ classifications are stable against quartic contact interactions, whereas the $\mathbb{Z}$ classifications reduce to ${\mathbb{Z}}_{N}$, where $N$ depends on the symmetry class and the dimension $d$. These results are derived using a quantum nonlinear $\ensuremath{\sigma}$ model (QNLSM) that describes the effects of the quartic interactions on the boundary modes of the crystalline topological phases. We use Clifford algebra extensions to derive the target spaces of these QNLSMs in a unified way. The reduction pattern of the free-fermion classification then follows from the presence or absence of topological terms in the QNLSMs, which is determined by the homotopy group of the target spaces. We show that this derivation can be performed using either a complex fermion or a real Majorana representation of the crystalline topological phases and demonstrate that these two representations give consistent results. To illustrate the breakdown of the noninteracting classification, we present examples of crystalline topological insulators and superconductors in one, two, and three dimensions whose surface modes are unstable against interactions. For the three-dimensional example, we show that the reduction pattern obtained by the QNLSM method agrees with the one inferred from the stability analysis of the boundary modes using bosonization.

arXiv:1609.07469•Physical review. B./Physical review. B•May 4, 2017•34 citations

Real-space construction and classification for time-reversal symmetric crystalline superconductors in 2D interacting fermionic systems

Yiming Liu, Wei-Qiang Chen, Zheng‐Cheng Gu

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 $C_4 \times Z_2^T$ or $D_4 \times Z_2^T$ (plus fermion parity), the intrinsic sector has a $Z_4$ classification. The corresponding root phases generating this $Z_4$ 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 $Z_4$ spinless intrinsic phases for wallpaper groups $p4$, $p4m$, and $p4g$. We also find an additional $Z_2$ intrinsically interacting phase for spinless fermions with wallpaper group $pm$, 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.

arXiv:2602.20560•Feb 24, 2026•0 citations
While free fermion topological crystalline insulators have been largely classified, the analogous problem in the strongly interacting case has been only partially solved, and the relationship between the free and interacting classifications is not well understood. In this paper, we develop a characterization and classification of interacting, invertible fermionic topological phases in (2+12+12+1) dimensions with charge conservation, discrete magnetic translation, and MMM-fold point group rotation symmetries, which form the group Gf=U(1)f\ifmmode×\else\texttimes\fi\ensuremathϕ[Z2\ensuremath⋊ZM]{G}_{f}=\text{U}{(1)}^{f}{\ifmmode\times\else\texttimes\fi{}}_{\ensuremath{\phi}}[{\mathbb{Z}}^{2}\ensuremath{\rtimes}{\mathbb{Z}}_{M}]Gf​=U(1)f\ifmmode×\else\texttimes\fi\ensuremathϕ​[Z2\ensuremath⋊ZM​] for M=1,2,3,4,and6M=1,2,3,4,\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}6M=1,2,3,4,and6. \ensuremathϕ\ensuremath{\phi}\ensuremathϕ is the magnetic flux per unit cell. We derive a topological response theory in terms of background crystalline gauge fields, which gives a complete classification of different phases and a physical characterization in terms of quantized response to symmetry defects. We then derive the same classification in terms of a set of real-space invariants \ensuremathΘo\ifmmode±\else\textpm\fi{{\mathrm{\ensuremath{\Theta}}}_{\text{o}}^{\ifmmode\pm\else\textpm\fi{}}}\ensuremathΘo\ifmmode±\else\textpm\fi​ that can be obtained from ground state expectation values of suitable partial rotation operators. We explicitly relate these real-space invariants to the quantized coefficients in the topological response theory, and find the dependence of the invariants on the chiral central charge c\ensuremath−{c}_{\ensuremath{-}}c\ensuremath−​ of the invertible phase. Finally, when \ensuremathϕ=0,\ensuremath{\phi}=0,\ensuremathϕ=0, we derive an explicit map between the free and interacting classifications.
Topology is routinely used to understand the physics of electronic insulators. However, for strongly interacting electronic matter, such as Mott insulators, a comprehensive topological characterization is still lacking. When their ground state only contains short-range entanglement and does not break symmetries spontaneously, they generically realize crystalline fermionic symmetry-protected topological phases (cFSPTs), supporting gapless modes at the boundaries or at the lattice defects. Here, we provide an exhaustive classification of cFSPTs in two dimensions with U ( 1 ) charge-conservation and spinful time-reversal symmetries, namely, those generically present in spin-orbit coupled insulators, for any of the 17 wallpaper groups. It has been shown that the classification of cFSPTs can be understood from appropriate real-space decorations of lower-dimensional subspaces, and we expose how these relate to the Wyckoff positions of the lattice. We find that all nontrivial one-dimensional decorations require electronic interactions. Furthermore, we provide model Hamiltonians for various decorations, and discuss the signatures of cFSPTs. This classification paves the way to further explore topological interacting insulators, providing the backbone information in generic model systems and ultimately in experiments. Published by the American Physical Society 2024
Topology is routinely used to understand the physics of electronic insulators. However, for strongly interacting electronic matter, such as Mott insulators, a comprehensive topological characterization is still lacking. When their ground state only contains short-range entanglement and does not break symmetries spontaneously, they generically realize crystalline fermionic symmetry-protected topological phases (cFSPTs), supporting gapless modes at the boundaries or at the lattice defects. Here, we provide an exhaustive classification of cFSPTs in two dimensions with charge-conservation and spinful time-reversal symmetries, namely, those generically present in spin-orbit coupled insulators, for any of the 17 wallpaper groups. It has been shown that the classification of cFSPTs can be understood from appropriate real-space decorations of lower-dimensional subspaces, and we expose how these relate to the Wyckoff positions of the lattice. We find that all nontrivial one-dimensional decorations require electronic interactions. Furthermore, we provide model Hamiltonians for various decorations, and discuss the signatures of cFSPTs. This classification paves the way to further explore topological interacting insulators, providing the backbone information in generic model systems and ultimately in experiments.
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 (n1,n2,n3,ν4)(n_1, n_2, n_3, ν_4)(n1​,n2​,n3​,ν4​) characterize a 3D FSPT with symmetry Gb×ω2Z2fG_b\times_{ω_2}Z_2^fGb​×ω2​​Z2f​, corresponding to p+ipp+ipp+ip, Kitaev chain, complex fermion, and bosonic SPT layers. Inspired by previous works, a crucial aspect is the p+ipp+ipp+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+ipp+ipp+ip superconductor, while the latter is determined by the free part of H1(Gb,ZT)H^1(G_b, Z_T)H1(Gb​,ZT​), 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 (n2,n3,ν4)(n_2, n_3, ν_4)(n2​,n3​,ν4​), which are amenable to automatic computation. As an application, we classify the 230 space-group topological crystalline superconductors in interacting electronic systems.
Crystalline symmetries give rise to topological invariants that can distinguish quantum phases of matter. Understanding these in strongly interacting systems is an ongoing research direction requiring non-perturbative methods. Recent developments have demonstrated that even classic models, like the Harper-Hofstadter model of free fermions on a lattice in a magnetic field, yield a host of crystalline symmetry protected topological invariants. Here we review some of these developments, focusing mainly on how to characterize, classify, and detect invariants arising from lattice translation and rotation symmetries along with charge conservation in two-dimensional systems, including integer and fractional Chern insulators.
We classify interacting topological insulators and superconductors with order-two crystal symmetries (reflection or twofold rotation), focusing on the case where interactions reduce the noninteracting classification. We find that the free-fermion Z2{\mathbb{Z}}_{2}Z2​ classifications are stable against quartic contact interactions, whereas the Z\mathbb{Z}Z classifications reduce to ZN{\mathbb{Z}}_{N}ZN​, where NNN depends on the symmetry class and the dimension ddd. These results are derived using a quantum nonlinear $\ensuremath{\sigma}$ model (QNLSM) that describes the effects of the quartic interactions on the boundary modes of the crystalline topological phases. We use Clifford algebra extensions to derive the target spaces of these QNLSMs in a unified way. The reduction pattern of the free-fermion classification then follows from the presence or absence of topological terms in the QNLSMs, which is determined by the homotopy group of the target spaces. We show that this derivation can be performed using either a complex fermion or a real Majorana representation of the crystalline topological phases and demonstrate that these two representations give consistent results. To illustrate the breakdown of the noninteracting classification, we present examples of crystalline topological insulators and superconductors in one, two, and three dimensions whose surface modes are unstable against interactions. For the three-dimensional example, we show that the reduction pattern obtained by the QNLSM method agrees with the one inferred from the stability analysis of the boundary modes using bosonization.
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.