We explore an exact duality in $(2+1)$ dimensions $[(2+1)\mathrm{D}]$ between the fermionization of a bosonic theory with a ${\mathbb{Z}}_{2}$ subsystem symmetry and a fermionic theory with a ${\mathbb{Z}}_{2}$ subsystem fermion parity symmetry. A typical example is the duality between the fermionization of the plaquette Ising model and the plaquette fermion model. We first revisit the standard boson-fermion duality in $(1+1)$ dimensions with a ${\mathbb{Z}}_{2}$ 0-form symmetry, presenting in a way generalizable to $(2+1)\mathrm{D}$ . We proceed to $(2+1)\mathrm{D}$ with a ${\mathbb{Z}}_{2}$ subsystem symmetry and establish the exact duality on the lattice by using the generalized Jordan-Wigner map, with a careful discussion on the mapping of the twist and symmetry sectors. This motivates us to introduce the subsystem Arf invariant, which exhibits a foliation structure.We explore an exact duality in $(2+1)$ dimensions $[(2+1)\mathrm{D}]$ between the fermionization of a bosonic theory with a ${\mathbb{Z}}_{2}$ subsystem symmetry and a fermionic theory with a ${\mathbb{Z}}_{2}$ subsystem fermion parity symmetry. A typical example is the duality between the fermionization of the plaquette Ising model and the plaquette fermion model. We first revisit the standard boson-fermion duality in $(1+1)$ dimensions with a ${\mathbb{Z}}_{2}$ 0-form symmetry, presenting in a way generalizable to $(2+1)\mathrm{D}$. We proceed to $(2+1)\mathrm{D}$ with a ${\mathbb{Z}}_{2}$ subsystem symmetry and establish the exact duality on the lattice by using the generalized Jordan-Wigner map, with a careful discussion on the mapping of the twist and symmetry sectors. This motivates us to introduce the subsystem Arf invariant, which exhibits a foliation structure.

arXiv:2206.02727Physical review. B./Physical review. BAug 29, 202223 citations

On the Relation between Fermions and Bosons

Morikazu TODA

Total mean curvature and submanifolds of finite typeOct 1, 19930 citations

Fermion decoration construction of symmetry-protected trivial order for fermion systems with any symmetry and in any dimension

Tian Lan, Chenchang Zhu, Xiao-Gang Wen

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 ${G}_{f}$ , which can be a nontrivial ${Z}_{2}^{f}$ extension ${Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}$ (where ${Z}_{2}^{f}$ is the fermion-number-parity symmetry and ${G}_{b}$ is the bosonic symmetry). This generalizes the previous results from group supercohomology of Gu and Wen (arXiv:1201.2648), where ${G}_{f}$ is assumed to be ${Z}_{2}^{f}\ifmmode\times\else\texttimes\fi{}{G}_{b}$. We find that the $(d+1)$ -dimensional $[(d+1)\mathrm{D}]$ fermionic SPT phases with bosonic symmetry ${G}_{b}$ 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 $(d+1)\mathrm{D}$ 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 $\mathcal{B}({Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}{O}_{\ensuremath{\infty}})\stackrel{\ensuremath{\varphi}}{\ensuremath{\rightarrow}}\mathcal{B}({O}_{\ensuremath{\infty}},1;{Z}_{2},d)]$.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 ${G}_{f}$, which can be a nontrivial ${Z}_{2}^{f}$ extension ${Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}$ (where ${Z}_{2}^{f}$ is the fermion-number-parity symmetry and ${G}_{b}$ is the bosonic symmetry). This generalizes the previous results from group supercohomology of Gu and Wen (arXiv:1201.2648), where ${G}_{f}$ is assumed to be ${Z}_{2}^{f}\ifmmode\times\else\texttimes\fi{}{G}_{b}$. We find that the $(d+1)$-dimensional $[(d+1)\mathrm{D}]$ fermionic SPT phases with bosonic symmetry ${G}_{b}$ 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 $(d+1)\mathrm{D}$ 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 $\mathcal{B}({Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}{O}_{\ensuremath{\infty}})\stackrel{\ensuremath{\varphi}}{\ensuremath{\rightarrow}}\mathcal{B}({O}_{\ensuremath{\infty}},1;{Z}_{2},d)]$.

arXiv:1809.01112Physical review. B./Physical review. BDec 26, 201928 citations

On the Relation between Fermions and Bosons

Morikazu Toda

Journal of the Physical Society of JapanMar 1, 19526 citations

Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries

Tian Lan, Liang Kong, Xiao-Gang Wen

We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry $G$ . The key is to use the so-called symmetric fusion category $\mathcal{E}$ to describe the symmetry. Here, $\mathcal{E}=\mathrm{sRep}({\mathbb{Z}}_{2}^{f})$ describing particles in a fermionic product state without symmetry, or $\mathcal{E}=\mathrm{sRep}({G}^{f}) [\mathcal{E}=\mathrm{Rep}(G)]$ describing particles in a fermionic (bosonic) product state with symmetry $G$ . Then, topological orders with symmetry $\mathcal{E}$ are classified by nondegenerate unitary braided fusion categories over $\mathcal{E}$ , plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to $p+\phantom{\rule{1.0pt}{0ex}}\mathrm{i}\phantom{\rule{1.0pt}{0ex}}p$ fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the $K=\left(\begin{array}{cc}\hfill \ensuremath{-}1& \hfill 0\\ \hfill 0& \hfill 2\end{array}\right)$ fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge $c=\frac{1}{4}$ , whose only topological excitation has non-Abelian statistics with spin $s=\frac{1}{4}$ and quantum dimension $d=1+\sqrt{2}$ .We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry $G$. The key is to use the so-called symmetric fusion category $\mathcal{E}$ to describe the symmetry. Here, $\mathcal{E}=\mathrm{sRep}({\mathbb{Z}}_{2}^{f})$ describing particles in a fermionic product state without symmetry, or $\mathcal{E}=\mathrm{sRep}({G}^{f}) [\mathcal{E}=\mathrm{Rep}(G)]$ describing particles in a fermionic (bosonic) product state with symmetry $G$. Then, topological orders with symmetry $\mathcal{E}$ are classified by nondegenerate unitary braided fusion categories over $\mathcal{E}$, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to $p+\phantom{\rule{1.0pt}{0ex}}\mathrm{i}\phantom{\rule{1.0pt}{0ex}}p$ fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the $K=\left(\begin{array}{cc}\hfill \ensuremath{-}1& \hfill 0\\ \hfill 0& \hfill 2\end{array}\right)$ fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge $c=\frac{1}{4}$, whose only topological excitation has non-Abelian statistics with spin $s=\frac{1}{4}$ and quantum dimension $d=1+\sqrt{2}$.

arXiv:1507.04673Physical review. B./Physical review. BOct 10, 2016126 citations

Bosons and Fermions

Arnout Jozef Ceulemans

Springer eBooksJan 1, 20221 citation

We explore an exact duality in

(2+1)

dimensions

[(2+1)\mathrm{D}]

between the fermionization of a bosonic theory with a

{\mathbb{Z}}_{2}

subsystem symmetry and a fermionic theory with a

{\mathbb{Z}}_{2}

subsystem fermion parity symmetry. A typical example is the duality between the fermionization of the plaquette Ising model and the plaquette fermion model. We first revisit the standard boson-fermion duality in

(1+1)

dimensions with a

{\mathbb{Z}}_{2}

0-form symmetry, presenting in a way generalizable to

(2+1)\mathrm{D}

. We proceed to

(2+1)\mathrm{D}

with a

{\mathbb{Z}}_{2}

subsystem symmetry and establish the exact duality on the lattice by using the generalized Jordan-Wigner map, with a careful discussion on the mapping of the twist and symmetry sectors. This motivates us to introduce the subsystem Arf invariant, which exhibits a foliation structure.

{G}_{f}

, which can be a nontrivial

{Z}_{2}^{f}

extension ${Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}$ (where

{Z}_{2}^{f}

is the fermion-number-parity symmetry and

{G}_{b}

is the bosonic symmetry). This generalizes the previous results from group supercohomology of Gu and Wen (arXiv:1201.2648), where

{G}_{f}

is assumed to be ${Z}_{2}^{f}\ifmmode\times\else\texttimes\fi{}{G}_{b}$. We find that the

(d+1)

-dimensional

[(d+1)\mathrm{D}]

fermionic SPT phases with bosonic symmetry

{G}_{b}

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)\mathrm{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 $\mathcal{B}({Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}{O}_{\ensuremath{\infty}})\stackrel{\ensuremath{\varphi}}{\ensuremath{\rightarrow}}\mathcal{B}({O}_{\ensuremath{\infty}},1;{Z}_{2},d)]$.

We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry

G

. The key is to use the so-called symmetric fusion category

\mathcal{E}

to describe the symmetry. Here,

\mathcal{E}=\mathrm{sRep}({\mathbb{Z}}_{2}^{f})

describing particles in a fermionic product state without symmetry, or

\mathcal{E}=\mathrm{sRep}({G}^{f}) [\mathcal{E}=\mathrm{Rep}(G)]

describing particles in a fermionic (bosonic) product state with symmetry

G

. Then, topological orders with symmetry

\mathcal{E}

are classified by nondegenerate unitary braided fusion categories over

\mathcal{E}

, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to

p+\phantom{\rule{1.0pt}{0ex}}\mathrm{i}\phantom{\rule{1.0pt}{0ex}}p

fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the $K=\left(\begin{array}{cc}\hfill \ensuremath{-}1& \hfill 0\\ \hfill 0& \hfill 2\end{array}\right)$ fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge

c=\frac{1}{4}

, whose only topological excitation has non-Abelian statistics with spin

s=\frac{1}{4}

and quantum dimension

d=1+\sqrt{2}