Michael Schuler, LouisPaul Henry, YuanMing Lu, Andreas M. LĂ¤uchli
SciPost Phys. 14, 001 (2023) ·
published 11 January 2023

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Anyons in a topologically ordered phase can carry fractional quantum numbers with respect to the symmetry group of the considered system, one example being the fractional charge of the quasiparticles and quasiholes in the fractional quantum Hall effect. When such symmetryfractionalized anyons condense, the resulting phase must spontaneously break the symmetry and display a local order parameter. In this paper, we study the phase diagram and anyon condensation transitions of a $\mathbb{Z}_2$ topological order perturbed by Ising interactions in the Toric Code. The interplay between the global (``onsite'') Ising ($\mathbb{Z}_2$) symmetry and the lattice space group symmetries results in a nontrivial symmetry fractionalization class for the anyons, and is shown to lead to two characteristically different confined, symmetrybroken phases. To understand the anyon condensation transitions, we use the recently introduced critical torus energy spectrum technique to identify a line of emergent 2+1D XY* transitions ending at a finetuned (Ising$^2$)* critical point. We provide numerical evidence for the occurrence of two symmetry breaking patterns predicted by the specific symmetry fractionalization class of the condensed anyons in the explored phase diagram. In combination with largescale quantum Monte Carlo simulations we measure unusually large critical exponents $\eta$ for the scaling of the correlation function at the continuous anyon condensation transitions, and we further identify lines of (weakly) first order transitions in the phase diagram. As an important additional result, we discuss the phase diagram of a resulting 2+1D AshkinTeller model, where we demonstrate that torus spectroscopy is capable of identifying emergent XY/O(2) critical behaviour, thereby solving some longstanding open questions in the domain of the 3D AshkinTeller models. To establish the generality of our results, we propose a field theoretical description capturing the transition from a $\mathbb{Z}_2$ topological order to either $\mathbb{Z}_2$ symmetry broken phase, which is in excellent agreement with the numerical results.
SciPost Phys. 11, 024 (2021) ·
published 6 August 2021

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We propose and prove a family of generalized LiebSchultzMattis (LSM) theorems for symmetry protected topological (SPT) phases on boson/spin models in any dimensions. The "conventional" LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin1/2 particles per unit cell, forbids a symmetric shortrangeentangled ground state in such a system. Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a nontrivial SPT phase with anomalous boundary excitations. Depending on models, they can be either strong or "higherorder" crystalline SPT phases, characterized by nontrivial surface/hinge/corner states. Furthermore, given the symmetry group and the spatial assignment of fractional spins, we are able to determine all possible SPT phases for a symmetric ground state, using the real space construction for SPT phases based on the spectral sequence of cohomology theory. We provide examples in one, two and three spatial dimensions, and discuss possible physical realization of these SPT phases based on condensation of topological excitations in fractionalized phases.