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arXiv:2110.03008 · Physical Review Letters

Numerical Evidence for a Haagerup Conformal Field Theory

Tzu-Chen Huang, Ying-Hsuan Lin, Kantaro Ohmori, Yuji Tachikawa, Masaki Tezuka

Published Jun 10, 2022• 59 citations
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Abstract

We numerically study an anyon chain based on the Haagerup fusion category and find evidence that it leads in the long-distance limit to a conformal field theory whose central charge is ∼2. Fusion categories generalize the concept of finite group symmetries to noninvertible symmetry operations, and the Haagerup fusion category is the simplest one which comes from neither finite groups nor affine Lie algebras. As such, ours is the first example of conformal field theories which have truly exotic generalized symmetries. Basically the same result was independently obtained in the preceding Letter [Phys. Rev. Lett. 128, 231602 (2022)PRLTAO0031-900710.1103/PhysRevLett.128.231602].

DOI
10.1103/physrevlett.128.231603

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We numerically study an anyon chain based on the Haagerup fusion category and find evidence that it leads in the long-distance limit to a conformal field theory whose central charge is ∼2. Fusion categories generalize the concept of finite group symmetries to noninvertible symmetry operations, and the Haagerup fusion category is the simplest one which comes from neither finite groups nor affine Lie algebras. As such, ours is the first example of conformal field theories which have truly exotic generalized symmetries. Basically the same result was independently obtained in the preceding Letter [Phys. Rev. Lett. 128, 231602 (2022)PRLTAO0031-900710.1103/PhysRevLett.128.231602].
Given a critical quantum spin chain described by a conformal field theory (CFT) at long distances, it is crucial to understand the universal conformal data. One most important ingredient is the operator product expansion (OPE) coefficients, which describe how operators fuse into each other. It has been proposed in [Zou and Vidal, Phys. Rev. B 105, 125125 (2022)] that the OPE coefficients can be computed from overlaps of low-energy wave functions of the spin chain. In this paper, we establish that all conformal data including central charge, conformal dimensions, and OPE coefficients are encoded in the wave-function overlaps, with universal finite-size corrections that depend on the operator content of the cyclic orbifold CFT. Thus this method allows us to numerically compute all the conformal data based solely on the low-energy eigenstates. The predictions are verified in the Ising and XXZ model. As an application, we study the recently proposed Haagerup model built from the Haagerup fusion category. We find that the CFT has central charge $c\ensuremath{\approx}2.1$ and the lowest spin-1 operator in the twisted sector has scaling dimension $1<{\mathrm{\ensuremath{\Delta}}}_{J}\ensuremath{\le}1.4$.
We use the formalism of strange correlators to construct a critical classical lattice model in two dimensions with the Haagerup fusion category H_{3} as input data. We present compelling numerical evidence in the form of finite entanglement scaling to support a Haagerup conformal field theory (CFT) with central charge c=2. Generalized twisted CFT spectra are numerically obtained through exact diagonalization of the transfer matrix, and the conformal towers are separated in the spectra through their identification with the topological sectors. It is further argued that our model can be obtained through an orbifold procedure from a larger lattice model with input Z(H_{3}), which is the simplest modular tensor category that does not admit an algebraic construction. This provides a counterexample for the conjecture that all rational CFT can be constructed from standard methods.
We construct the first integrable models based on the Haagerup fusion category H 3 . We introduce a Haagerup version of the anyonic spin chain and use the boost operator formalism to identify two integrable Hamiltonians of PXP type on this chain. The first of these is an analog of the golden chain; it has a topological symmetry based on H 3 and satisfies the Temperley-Lieb algebra with parameter δ = ( 3 + 13 ) / 2 . We prove its integrability using a Lax formalism, and construct the corresponding solution to the Yang-Baxter equation. We present numerical evidence that this model is gapless with a dynamical critical exponent z ≠ 1 . The second integrable model we find breaks the topological symmetry. We present numerical evidence that this model reduces to a CFT in the large volume limit with central charge c ∼ 3 / 2 .
Using the numerical modular bootstrap, we constrain the space of 1+1d1+1\mathrm{d}1+1d conformal field theories (CFTs) with a finite noninvertible global symmetry described by a fusion category C\mathcal{C}C. We derive universal and rigorous upper bounds on the lightest C\mathcal{C}C-preserving scalar local operator for fusion categories such as the Ising and Fibonacci categories. These numerical bounds constrain the possible robust gapless phases protected by a noninvertible global symmetry, which commonly arise from microscopic lattice models such as the anyonic chains. We also derive bounds on the lightest C\mathcal{C}C-violating local operator. Our bootstrap equations naturally arise from a ``slab construction,'' where the CFT is coupled to the 2+1d2+1\mathrm{d}2+1d Turaev-Viro topological quantum field theory , also known as the symmetry TFT.
We construct a (1+1)d gapless theory which has Haagerup H_{3} symmetry. The construction relies on the recent exploration of the categorical Landau paradigm applied to fusion category symmetries. First, using the symmetry topological field theory, we construct all gapped phases with Haagerup symmetry. Extending this construction to gapless phases, we study the second order phase transition between gapped phases, and determine analytically a Haagerup-symmetric conformal field theory. This is given in terms of two copies of the three-state Potts model, on which we realize the full H_{3} symmetry action and determine the relevant deformations to the H_{3}-symmetric gapped phases. This continuum analysis is corroborated by a lattice model construction of the gapped and gapless phases, using the anyon chain.
We study generalized symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. This paper follows our companion paper on gapped phases and anomalies associated with these symmetries. In the present work we focus on identifying fusion category symmetries, using both specialized 1+1D methods such as the modular bootstrap and (rational) conformal field theory (CFT), as well as general methods based on gauging finite symmetries, that extend to all dimensions. We apply these methods to c=1c = 1c=1 CFTs and uncover a rich structure. We find that even those c=1c = 1c=1 CFTs with only finite group-like symmetries can have continuous fusion category symmetries, and prove a Noether theorem that relates such symmetries in general to non-local conserved currents. We also use these symmetries to derive new constraints on RG flows between 1+1D CFTs.