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arXiv:2303.16919 · Physical Review Letters

Complete Crystalline Topological Invariants from Partial Rotations in ( 2 + 1 ) D Invertible Fermionic States and Hofstadter’s Butterfly

Yuxuan Zhang, Naren Manjunath, Ryohei Kobayashi, Maissam Barkeshli

Published Oct 23, 2023• 17 citations
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Abstract

The theory of topological phases of matter predicts invariants protected only by crystalline symmetry, yet it has been unclear how to extract these from microscopic calculations in general. Here, we show how to extract a set of many-body invariants ${{\mathrm{\ensuremath{\Theta}}}_{o}^{\ifmmode\pm\else\textpm\fi{}}}$, where $o$ is a high symmetry point, from partial rotations in $(2+1)\mathrm{D}$ invertible fermionic states. Our results apply in the presence of magnetic field and Chern number $C\ensuremath{\ne}0$, in contrast to previous work. ${{\mathrm{\ensuremath{\Theta}}}_{o}^{\ifmmode\pm\else\textpm\fi{}}}$ together with $C$, chiral central charge ${c}_{\ensuremath{-}}$, and filling $\ensuremath{\nu}$ provide a complete many-body characterization of the topological state with symmetry group $G=U(1){\ifmmode\times\else\texttimes\fi{}}_{\ensuremath{\phi}}[{\mathbb{Z}}^{2}\ensuremath{\rtimes}{\mathbb{Z}}_{M}]$. Moreover, all these many-body invariants can be obtained from a single bulk ground state, without inserting additional defects. We perform numerical computations on the square lattice Hofstadter model. Remarkably, these match calculations from conformal and topological field theory, where $G$-crossed modular $S$, $T$ matrices of symmetry defects play a crucial role. Our results provide additional colorings of Hofstadter's butterfly, extending recently discovered colorings by the discrete shift and quantized charge polarization.

DOI
10.1103/physrevlett.131.176501

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The theory of topological phases of matter predicts invariants protected only by crystalline symmetry, yet it has been unclear how to extract these from microscopic calculations in general. Here, we show how to extract a set of many-body invariants \ensuremathΘo\ifmmode±\else\textpm\fi{{\mathrm{\ensuremath{\Theta}}}_{o}^{\ifmmode\pm\else\textpm\fi{}}}\ensuremathΘo\ifmmode±\else\textpm\fi​, where ooo is a high symmetry point, from partial rotations in (2+1)D(2+1)\mathrm{D}(2+1)D invertible fermionic states. Our results apply in the presence of magnetic field and Chern number C\ensuremath≠0C\ensuremath{\ne}0C\ensuremath=0, in contrast to previous work. \ensuremathΘo\ifmmode±\else\textpm\fi{{\mathrm{\ensuremath{\Theta}}}_{o}^{\ifmmode\pm\else\textpm\fi{}}}\ensuremathΘo\ifmmode±\else\textpm\fi​ together with CCC, chiral central charge c\ensuremath−{c}_{\ensuremath{-}}c\ensuremath−​, and filling \ensuremathν\ensuremath{\nu}\ensuremathν provide a complete many-body characterization of the topological state with symmetry group G=U(1)\ifmmode×\else\texttimes\fi\ensuremathϕ[Z2\ensuremath⋊ZM]G=U(1){\ifmmode\times\else\texttimes\fi{}}_{\ensuremath{\phi}}[{\mathbb{Z}}^{2}\ensuremath{\rtimes}{\mathbb{Z}}_{M}]G=U(1)\ifmmode×\else\texttimes\fi\ensuremathϕ​[Z2\ensuremath⋊ZM​]. Moreover, all these many-body invariants can be obtained from a single bulk ground state, without inserting additional defects. We perform numerical computations on the square lattice Hofstadter model. Remarkably, these match calculations from conformal and topological field theory, where GGG-crossed modular SSS, TTT matrices of symmetry defects play a crucial role. Our results provide additional colorings of Hofstadter's butterfly, extending recently discovered colorings by the discrete shift and quantized charge polarization.
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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 ${G}_{f}=\text{U}{(1)}^{f}{\ifmmode\times\else\texttimes\fi{}}_{\ensuremath{\phi}}[{\mathbb{Z}}^{2}\ensuremath{\rtimes}{\mathbb{Z}}_{M}]$ for M=1,2,3,4,and6M=1,2,3,4,\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}6M=1,2,3,4,and6. $\ensuremath{\phi}$ 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.
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.
In the presence of crystalline symmetry, topologically ordered states can acquire a host of symmetry-protected invariants. These determine the patterns of crystalline symmetry fractionalization of the anyons in addition to fractionally quantized responses to lattice defects. Here we show how ground state expectation values of partial rotations centered at high symmetry points can be used to extract crystalline invariants. Using methods from conformal field theory and G-crossed braided tensor categories, we develop a theory of invariants obtained from partial rotations, which apply to both Abelian and non-Abelian topological orders. We then perform numerical Monte Carlo calculations for projected parton wave functions of fractional Chern insulators, demonstrating remarkable agreement between theory and numerics. For the topological orders we consider, we show that the Hall conductivity, filling fraction, and partial rotation invariants fully characterize the crystalline invariants of the system. Our results also yield invariants of continuum fractional quantum Hall states protected by spatial rotational symmetry.
Recently, it has been shown how topological phases of matter with crystalline symmetry and U(1)U(1)U(1) charge conservation can be partially characterized by a set of many-body invariants, the discrete shift So{\mathcal{S}}_{\mathrm{o}}So​ and electric polarization ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$, where o\mathrm{o}o labels a high-symmetry point. Crucially, these can be defined even with nonzero Chern number and/or magnetic field. One manifestation of these invariants is through quantized fractional contributions to the charge in the vicinity of a lattice disclination or dislocation. In this paper, we show that these invariants can also be extracted from the length and corner dependence of the total charge (mod 1) on the boundary of the system. We provide a general formula in terms of So{\mathcal{S}}_{\mathrm{o}}So​ and ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$ for the total charge of any subregion of the system which can include full boundaries or bulk lattice defects, unifying boundary, corner, disclination, and dislocation charge responses into a single general theory. These results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently. We also discuss how our theory can fully characterize the topological response of quadrupole insulators.
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