Constructing quantum localization operators using conformal symmetry

L. D. Faddeev

Ashley Milsted, Guifré Vidal

We study the emergence of 2D conformal symmetry in critical quantum spin chains on the finite circle. Our goal is to characterize the conformal field theory (CFT) describing the universality class of the corresponding quantum phase transition. Using only the lattice Hamiltonian $H = \sum_j h_j$ as an input, we construct operators $H_n$ (Fourier modes of the local Hamiltonian terms $h_j$) that transform the low-energy eigenstates of $H$ in the same way as certain combinations of the Virasoro generators do in the CFT. These operators were introduced for certain integrable lattice models by Koo and Saleur [Nucl. Phys. B 426, 459 (1994)]. In this paper we propose and demonstrate the use of operators $H_n$ to extract conformal data in nonintegrable quantum spin chains. In particular, we explain how to estimate the central charge from a simple ground-state expectation value and to systematically identify the low-energy states of $H$ corresponding to Virasoro primary operators, as must be done in order to identify the CFT.

Ying-Hsuan Lin, Shu-Heng Shao

Using the numerical modular bootstrap, we constrain the space of $1+1\mathrm{d}$ conformal field theories (CFTs) with a finite noninvertible global symmetry described by a fusion category $\mathcal{C}$. We derive universal and rigorous upper bounds on the lightest $\mathcal{C}$-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 $\mathcal{C}$-violating local operator. Our bootstrap equations naturally arise from a ``slab construction,'' where the CFT is coupled to the $2+1\mathrm{d}$ Turaev-Viro topological quantum field theory , also known as the symmetry TFT.

L. D. Faddeev

Zbigniew Haba

Stathis Vitouladitis

We establish a state-operator correspondence for a class of non-conformal quantum field theories with continuous higher-form symmetries and a mixed anomaly. Such systems can always be realised as a relativistic superfluid. The symmetry structure induces an infinite tower of conserved charges, which we construct explicitly. These charges satisfy an abelian current algebra with a central extension, generalising the familiar Kac–Moody algebras to higher dimensions. States and operators are organised into representations of this algebra, enabling a direct correspondence. We demonstrate the correspondence explicitly in free examples by performing the Euclidean path integral on a d d -dimensional ball, with local operators inserted in the origin, and matching to energy eigenstates on S^{d-1} S d − 1 obtained by canonical quantisation. Interestingly, in the absence of conformal invariance, the empty path integral prepares a squeezed vacuum rather than the true ground state.