Session D17: Topological Spin Liquids

From Foliated to Exotic Field Theories for Gapped Fracton Phases

Fracton phases of matter are gapped phases of matter that, by dint of their sensitivity to UV data, demand non-standard quantum field theories to describe them in the IR. Two such approaches are foliated quantum theory and exotic field theory. In this talk, we briefly introduce both types of field theory and outliine the procedure to obtain one from the other. This procedure recovers the equivalence between the foliated and exotic fractonic BF theories recently demonstrated at the level of operator correspondence. It can also be used demonstrate the equivalence of toric code layers and the anisotropic model with lineons and planons to the foliated BF theory with one and two foliations, respectively. Finally, it can be used to derive new exotic field theories that provide simple descriptions of hybrid fracton phases from foliated field theries known to do so. Our results both provide new examples of exotic field theories and pave the way toward their systematic construction from foliated field theories.   

Measuring Topological Field Theories: Lattice Models and Field-Theoretic Description

Recent years have witnessed a surge of interest in performing measurements within topological phases of matter, e.g., symmetry-protected topological (SPT) phases and topological orders. Notably, measurements of certain SPT states have been known to be related to Kramers-Wannier duality and Jordan-Wigner transformations, giving rise to long-range entangled states and invertible phases such as the Kitaev chain. Simultaneously, measurements of topologically ordered states correspond to charge condensations. In this work, we present a field-theoretic framework for the description of measurements within topological field theories. We employ various lattice models as examples to illustrate the outcomes of measuring local symmetry operators within topological phases, demonstrating their agreement with the predictions from field-theoretic descriptions. We demonstrate that these measurements lead to SPT phases, symmetry spontaneously breaking phases, and topologically ordered phases. Specifically, when there is emergent symmetry after measurement, the remaining symmetry and emergent symmetry will have a mixed anomaly, which lead to long-ranged entanglement.   

Anyon Condensation Web in 2D Modulated Gauge Theories

We introduce an anyon condensation web that interconnects a broad class of two dimensional spatially modulated gauge theories at a microscopic level. We find that condensation of anyons triggers the emergence of additional modulated symmetries, which has the general effect of increasing the number of super-selection anyon sectors, as well as the ground state degeneracy for systems with periodic boundary conditions. With this, we manage to connect several well known fracton-like states in exactly solvable ZN models as different phases of local Hamiltonians.   

Unambiguous diagnostic of chirality in the entanglement spectrum of (2+1)d topological spin liquids: Demonstration in a SU(4) PEPS

We address the key question of representation of chiral topological quantum states in (2+1)d by Projected Entangled Pair States (PEPS). A noted result (due to Wahl, Tu, Schuch, and Cirac [arXiv:1308.0316] and Dubail and Read [arXiv:1307.7726]) says this is possible for non-interacting fermions, but the answer is as yet unknown for interacting systems. Characteristic counting of degeneracies of low-lying states in the entanglement spectrum (ES) at fixed transverse momentum of bipartitioned long cylinders ("Li-Haldane counting") provides often-used supporting evidence for chirality. However, non-chiral states (with zero chiral central charge), yet strongly breaking time-reversal and reflection symmetries (i.e., "apparently" chiral states), are known [arXiv:2207.03246], whose low-lying ES exhibits the same Li-Haldane counting as a chiral state (with non-zero chiral charge) in certain topological sectors. We recently identified [arXiv:2305.13240] a hallmark of chirality of a wave function in the ES in SU(3) chiral spin liquids, which is a finer diagnostic than Li-Haldane counting: The exact degeneracies of entanglement-energy levels in the ES corresponding to paired conjugate representations, which are split in non-chiral states. Here we identify an analogous hallmark of chirality of wave functions of SU(4) spin liquids and demonstrate its use in a SU(4) PEPS.   

Eigenstate switching of topologically ordered states using local non-Hermitian perturbations

Topologically ordered phases have robust degenerate ground states against the local perturbations, thus it provides a promising platform for fault-tolerant quantum computation. While the non-local storage of information ensures robustness against local perturbations, it is unclear how to manipulate such an order in a versatile manner. In this work, we explore the potential of non-Hermitian exceptional points (EP) in the control of topological order. EPs arise due to non-Hermiticity, leading to the coalescence of eigenstates. We propose a novel scheme that utilizes $mathcal{PT}$-symmetric perturbations to induce eigenstate coalescence between different ground states of the toric code model. Adiabatic encircling EPs allows for the controlled switching of eigenstates, enabling dynamic manipulation of the topological degeneracy. Interestingly, we show a remarkable property of our scheme that arbitrary strengths of local perturbations can induce the EP and eigenstate switching. Finally, in a non-adiabatic regime, we also show the orientation-dependent behavior of non-adiabatic transitions (NAT) during EP encirclement. our work that controls the non-Hermiticity can serve as a promising strategy for fault-tolerant quantum information processing.   

Fractons, Strange Correlators and Dualities via Cluster State Measurements

The Calderbank-Shor-Steane (CSS) code is a class of quantum error correction codes that contains the toric code and fracton models. We use the chain complex representation of CSS codes to arrive at a web of dualities that generalizes the Wegner's duality in generalized quantum Ising models. Further, we provide a procedure to obtain statistical models associated with a given CSS code via the foliated cluster state, and derive a generalization of the Kramers-Wannier-Wegner duality for such statistical models. Moreover, we give a measurement-assisted gauging method with cluster-state entanglers for CSS/fracton models, extending results in the literature. We analyze the non-invertible fusion of the Kramers-Wannier transformation operators underpinning the gauging procedure. The cluster-state entanglers also allow us to construct a generalization of the notion of strange correlators to fracton models.   

Replica topological order and quantum error correction

Topological phases of matter offer a promising platform for quantum computation and quantum error correction. Nevertheless, unlike its counterpart in pure states, descriptions of topological order in mixed states remain relatively underexplored. Our study gives various equivalent definitions for replica topological order in mixed states. Similar to the replica trick, our definitions also involve n copies of density matrices of the mixed state. Using our framework, we can categorize topological orders in mixed states as either quantum or classical, depending on which type of information they encode. For the case of the toric code (TC) model in the presence of decoherence, we associate for each phase a quantum channel and describe the structure of the density matrix space. We show that in the quantum-topological phase, there exists a postselection-based error correction protocol that recovers the quantum information. We accomplish this using tensor networks by identifying boundary symmetry-protected topological phases to bulk anyon condensations via a bulk-boundary correspondence. As a result, we argue that the phase boundaries of the quantum replica topological mixed states correspond to the error threshold in the toric code with postselection. We anticipate our findings to be consistent in the analytical continuation as n → 1.