Pseudo-Hermitian Levin–Wen models from non-semisimple TQFTs

Geer, Nathan; Lauda, Aaron D.; Patureau-Mirand, Bertrand; Sussan, Joshua

doi:10.1016/j.aop.2022.168937

Cited by 3 publications

(12 citation statements)

References 25 publications

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“…The next result follows easily from the previous proposition and is a generalization of [24, Corollary III.3]. Corollary Let

p

and

p^{\prime }

be two plaquettes.…”

Section: Relative G$\mathcal {G}$‐lw‐systemmentioning

confidence: 68%

“…\end{equation}$$Thus, a Hamiltonian, or Hermitian operator

H \colon \mathcal {H} \rightarrow \mathcal {H}

with respect to the inner product

\langle,\rangle

, then becomes pseudo‐Hermitian with respect to the inner product

\langle -,-\rangle _+

. See [24, Section II.C] for more details.…”

Section: Relative G$\mathcal {G}$‐lw‐systemmentioning

confidence: 99%

“…Define a new inner product on

\mathcal {H}=\mathcal {H}(\Gamma,{\Phi })

via

\begin{equation} \langle \cdot \mid \cdot \rangle:= \langle \cdot \mid \eta ^{-1}(\cdot) \rangle _+, \quad \quad \langle \Gamma, \sigma \mid \Gamma, \sigma ^{\prime } \rangle = \; \frac{ \prod _{v\in \Gamma _0}\gamma (\sigma (v))\prod _{e\in \Gamma _1}\beta (\sigma (e)) }{ \prod _{e\in \Gamma _1} \operatorname{\mathsf {d}}(\sigma (e)) }\ \cdot \delta _{\sigma, \sigma ^{\prime }}. \end{equation}

It is not difficult to show that this new pairing is Hermitian (see [24, Section II.C]). However, it is in general indefinite (see [24, Sections II.C and V]).…”

Section: Relative G$\mathcal {G}$‐lw‐systemmentioning

confidence: 99%

“…\end{equation}$$It is not difficult to show that this new pairing is Hermitian (see [24, Section II.C]). However, it is in general indefinite (see [24, Sections II.C and V]). In what follows, for an operator

F

, we let

F^{\dagger }

be the adjoint with respect to this indefinite form.…”

Section: Relative G$\mathcal {G}$‐lw‐systemmentioning

confidence: 99%

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Non‐semisimple Levin–Wen models and Hermitian TQFTs from quantum (super)groups

Geer,

Lauda,

Patureau‐Mirand

et al. 2024

Journal of London Math Soc

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We develop the categorical context for defining Hermitian non‐semisimple topological quantum field theories (TQFTs). We prove that relative Hermitian modular categories give rise to modified Hermitian Witten–Reshetikhin–Turaev TQFTs and provide numerous examples of these structures coming from the representation theory of quantum groups and quantum superalgebras. The Hermitian theory developed here for the modified Turaev–Viro TQFT is applied to define new pseudo‐Hermitian topological phases that can be considered as non‐semisimple analogs of Levin–Wen models.

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“…The next result follows easily from the previous proposition and is a generalization of [24, Corollary III.3]. Corollary Let

p

and

p^{\prime }

be two plaquettes.…”

Section: Relative G$\mathcal {G}$‐lw‐systemmentioning

confidence: 68%

“…\end{equation}$$Thus, a Hamiltonian, or Hermitian operator

H \colon \mathcal {H} \rightarrow \mathcal {H}

with respect to the inner product

\langle,\rangle

, then becomes pseudo‐Hermitian with respect to the inner product

\langle -,-\rangle _+

. See [24, Section II.C] for more details.…”

Section: Relative G$\mathcal {G}$‐lw‐systemmentioning

confidence: 99%

“…Define a new inner product on

\mathcal {H}=\mathcal {H}(\Gamma,{\Phi })

via

\begin{equation} \langle \cdot \mid \cdot \rangle:= \langle \cdot \mid \eta ^{-1}(\cdot) \rangle _+, \quad \quad \langle \Gamma, \sigma \mid \Gamma, \sigma ^{\prime } \rangle = \; \frac{ \prod _{v\in \Gamma _0}\gamma (\sigma (v))\prod _{e\in \Gamma _1}\beta (\sigma (e)) }{ \prod _{e\in \Gamma _1} \operatorname{\mathsf {d}}(\sigma (e)) }\ \cdot \delta _{\sigma, \sigma ^{\prime }}. \end{equation}

It is not difficult to show that this new pairing is Hermitian (see [24, Section II.C]). However, it is in general indefinite (see [24, Sections II.C and V]).…”

Section: Relative G$\mathcal {G}$‐lw‐systemmentioning

confidence: 99%

F

, we let

F^{\dagger }

be the adjoint with respect to this indefinite form.…”

Section: Relative G$\mathcal {G}$‐lw‐systemmentioning

confidence: 99%

See 2 more Smart Citations

Non‐semisimple Levin–Wen models and Hermitian TQFTs from quantum (super)groups

Geer,

Lauda,

Patureau‐Mirand

et al. 2024

Journal of London Math Soc

Self Cite

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show abstract

“…In our previous work [27,26], we have advocated that the topological advantages of nonsemisimple TQFTs may translate into potential advantages for constructing quantum systems based on topological phases of matter. In [25], we studied non-semisimple TQFTs associated with unrolled sl 2 and defined a Hermitian structure equipping the tensor spaces with Hermitian forms of possibly mixed signature.…”

Section: Introductionmentioning

confidence: 99%

A Hermitian TQFT from a non-semisimple category of quantum $${\mathfrak {sl}(2)}$$-modules

et al. 2022

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We study the density of the Burau representation from the perspective of a nonsemisimple TQFT at a fourth root of unity. This gives a TQFT construction of Squier's Hermitian form on the Burau representation with possibly mixed signature. We prove that the image of the braid group in the space of possibly indefinite unitary representations is dense. We also argue for the potential applications of non-semisimple TQFTs toward topological quantum computation.

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Internal Levin-Wen models

Mulevičius,

Runkel,

Voß

2024

SciPost Phys.

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Levin–Wen models are a class of two-dimensional lattice spin models with a Hamiltonian that is a sum of commuting projectors, which describe topological phases of matter related to Drinfeld centres. We generalise this construction to lattice systems internal to a topological phase described by an arbitrary modular fusion category \mathcal{C}𝒞. The lattice system is defined in terms of an orbifold datum \mathbb{A}𝔸 in \mathcal{C}𝒞, from which we construct a state space and a commuting-projector Hamiltonian H_{\mathbb{A}}H𝔸 acting on it. The topological phase of the degenerate ground states of H_{\mathbb{A}}H𝔸 is characterised by a modular fusion category \mathcal{C}_\mathbb{A}𝒞𝔸 defined directly in terms of \mathbb{A}𝔸. By choosing different \mathbb{A}𝔸’s for a fixed \mathcal{C}𝒞, one obtains precisely all phases which are Witt-equivalent to \mathcal{C}𝒞. As special cases we recover the Kitaev and the Levin–Wen lattice models from instances of orbifold data in the trivial modular fusion category of vector spaces, as well as phases obtained by anyon condensation in a given phase \mathcal{C}𝒞.

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Pseudo-Hermitian Levin–Wen models from non-semisimple TQFTs

Cited by 3 publications

References 25 publications

Non‐semisimple Levin–Wen models and Hermitian TQFTs from quantum (super)groups

Non‐semisimple Levin–Wen models and Hermitian TQFTs from quantum (super)groups

A Hermitian TQFT from a non-semisimple category of quantum $${\mathfrak {sl}(2)}$$-modules

Internal Levin-Wen models

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