TSVP Symposium: Aspects of Generalized Symmetries

In fault-tolerant quantum computing with the surface code, non-Clifford gates are crucial for universal computation. However, implementing these gates using methods like magic state distillation and code switching requires significant resources. In this talk, I will talk about a new protocol to realize logical non-Clifford operations with the potential for fault tolerance. Our approach begins with a special logical state in the Z4 surface code. By applying a sequence of transformations given by gauging a global symmetry and anyon condensation, the system goes through different topological codes, including the non-Abelian D4 quantum double model. This process ultimately produces a magic state in a condensed Z2 surface code, which enables the implementation of a logical T gate in the standard Z2 surface code. This work addresses the challenge of achieving universal fault-tolerant quantum computation in strictly two spatial dimensions and introduces a new approach that leverages non-Abelian topological phases. This talk will be based on arXiv:2502.00998 with Yanzhu Chen.

Fracton topological phases now have diverse research interests,　
involving various branches of physics, such as condensed matter physics,
and high-energy physics. Distinctive feature of these phases is that
mobility constraints are imposed on fractionalized quasi-particle
excitations (i.e., anyons), giving rise to sub-extensive ground state
degeneracy on a torus geometry. Due to this property, one faces an issue
with the UV/IR mixing, implying incapability to resort to preexisting
theoretical frameworks, such as topological field theories.

In this talk, we discuss a new type of symmetry -- modulated symmetry,
associated with conservation of multipoles (e.g, dipole) as well as
global charges. We propose BF theories, consisting of gauge fields
related to such a symmetry and see how fracton-like topological phases can be
constructed.

We discuss the mechanism of "symmetry-from-anomaly" in condensed matter-related models in both 1d and 3d spaces (which correspond to (1+1)d and (3+1)d space-time). Within the models discussed here, we establish the connection between field theory quantities such as different versions of the axial charge, and quantities with simple physical meanings in our systems. In our models and likely a class of related constructions, the existence of a topological order is necessary for the purpose of properly defining the axial symmetry. But the proper axial symmetry we define, though requires a topological order, is different from the noninvertible axial symmetry discussed in recent proposals.

We systematically construct tensor network representations for a broad class of intrinsically mixed-state topological phases and demonstrate that they are renormalization fixed points. Examples include tensor networks for generic Abelian topological orders subject to various decoherence channels, certain decohered non-Abelian Kitaev quantum doubles, and chiral topological phases in mixed states.

One dimensional spin chains display a surprisingly rich array of possible phases. We establish a correspondence between three such phases: gapless SPTs with G symmetry, a subset of gapped SPTs with G x G symmetry, and mixed state SPTs. In doing so, we extend the framework of SymTFT to mixed states and explore some of its applications. 

Quantum spin ice (QSI) is a lattice spin-model realization of full-fledged quantum electrodynamics, including photons, electric charges, and magnetic monopoles [1]. As one of the most interesting quantum spin liquids, a significant amount of experimental and theoretical investigation has been done in this field. I will present an overview of the quantum spin ice physics and also discuss our recent ongoing work [2] on how, in the so-called dipole-octupole QSI [3], one can experimentally have clean control of the dynamics of its emergent QED, including the transition between different symmetry-enriched phases, tuning the dispersion of photons and fine-structure constants, etc. One of the most straightforward experiments can achieve this: turning on the external magnetic field in the right direction.  

[1] Experimental signatures of emergent quantum electrodynamics in Pr2Hf2O7, Romain Sibille, Nicolas Gauthier, HY et al., Nature Physics 14, 711–715 (2018)
[2] Experimentally tunable QED in dipolar-octupolar quantum spin ice. HY, Alaric Sanders, Claudio Castelnovo, Andriy H Nevidomskyy, arXiv:2312.11641. 
[3] Evidence for fractional matter coupled to an emergent gauge field in a quantum spin ice, Victor Porée, HY et al., Nature Physics 21, 83–88 (2025)

Quasiperiodic systems offer a unique platform for realizing exotic quantum phases that challenge conventional understanding of condensed matter phenomena. In this work, we explore the emergence of long-distance magnetic correlations, multipolar physics  and  pattern-selective superconductivity in quasiperiodic systems. By tuning quasiperiodicity parameters and interaction strengths, we identify the emergence of unexpectedly long-range magnetic correlations in the absence of translational symmetry. In addition, we also point out that quasicrystals could generically host the multipolar degrees of freedom. Furthermore, we demonstrate that superconducting pairing in quasiperiodic systems can exhibit strong spatial selectivity, favoring distinct local configurations — an effect rooted in the underlying aperiodic geometry. These findings shed light on the interplay between structural complexity and collective quantum behavior, opening new avenues for designing materials with unconventional magnetic and superconducting properties.

Fusion surface models extend the framework of anyon chains to 2+1 dimensions. Constructed from braided fusion categories, these lattice models exhibit categorical 1-form symmetries and provide systematic generalizations of Kitaev’s honeycomb model. Many of them are promising candidates for realizing chiral topological order. In this talk, I will discuss how the fusion surface model framework can be extended to accommodate dualities—understood as gauging of 1-form symmetries. The dual models can potentially exhibit independent 0-form symmetries and realize symmetry-enriched topological order. As an example, I will present a duality between a spin-1/2 model on the honeycomb lattice and a constrained Hilbert space model with a Rep(S3) 1-form symmetry. This balk is based on arXiv:2408.04006, 2501.14722.

I will discuss the interplay between dualities and exact quantum many-body scars, motivated by my recent papers  arXiv:2505.21921 and  arXiv:2501.12514.

Coupled-layer constructions are invaluable for constructing exactly-solvable models of phases of matter. For example, the 3+1d X-cube model, a prototypical example of fractons, can be constructed by starting with a stack of 2+1d toric codes and turning on a coupling which condenses a particular composite object called a "particle-string" or "p-string". In this talk, I will show that the p-string can be thought of as a symmetry defect of a topological 1-form symmetry and reinterpret p-string condensation as the gauging of this higher-form symmetry. This approach clarifies the symmetry principles underlying p-string condensation and generalizes the familiar connection between anyon condensation and one-form gauging in 2+1d. Finally, I will comment on a rich gauging web relating the X-cube model, the 3+1d toric code, and SPT phases protected by both subsystem and higher-form symmetries, which exhibit phenomena such as subsystem symmetry fractionalization, as well as non-trivial extensions between subsystem and topological symmetries.

Gauging is a systematic way to construct a model with non-invertible symmetry from a model with ordinary group symmetry. In 2+1 dimensions or higher, one can generalize the standard gauging procedure by stacking a symmetry-enriched topological order before gauging the symmetry. This generalized gauging procedure allows us to realize a large class of non-invertible symmetries. In this talk, I will describe the generalized gauging of finite group symmetries in 2+1d lattice models. By applying the generalized gauging to gapped phases with ordinary group symmetry, I will construct commuting projector models for gapped phases with non-invertible symmetry.

Models of interacting particles with conserved quantities beyond particle number can lead to strong constraints on the pathways of relaxation to equilibrium. This can lead to a qualitative change in the timescale associated with the relaxation of even more severe effects like glassiness, where the system fails to equilibrate at all. This has already been studied for one-dimensional spin chains and here we consider a simple two-dimensional model to investigate relaxation for slowly varying macroscopic structures such as density waves with a long wavelength. We find that we are able to tune smoothly between a glassy regime and one controlled by sub-diffusive behavior using just the macroscopic features of the initial state. This result shows that the time required before a hydrodynamic description kicks in can be anomalously large, leading to real systems which are practically never controlled by the expected hydrodynamic behavior.

Non-thermal energy eigenstates of thermalizing isolated quantum systems have been intensively studied these days, as counter examples for the strong eigenstate thermalization hypothesis. Nowadays, it is believed that, based on many numerical evidence, these exceptional non-thermal states emerge when the Hamiltonian has a relatively small invariant subspace, which weakly violate ergodicity in the Hilbert space.
In the talk, we demonstrate that weak violations of ergodicity can be achieved by embedding selected integrable models into the larger Hilbert spaces of otherwise chaotic systems. Interestingly, models constructed in this manner can be viewed as perturbations of systems exhibiting Hilbert space fragmentation, where the perturbations eliminate the fragmentation. Furthermore, we show that this approach to constructing weakly ergodicity-breaking models can also be extended to open quantum systems.

Coming soon...

We examine the noninvertible symmetry (NIS) in one-dimensional (1D) SPT models protected by dipolar and exponential-charge symmetries, which are the two examples of modulated SPT, or MSPT for short. To lay the ground, we first consider the NIS of the Z_N x Z_N cluster model, which is an example of SPT protected by two charge symmetries, while an earlier work considered the NIS of Z_2 x Z_2 cluster model. In all three models of SPT with charge, dipole, and exponential-charge symmetries, we explicitly work out the noninvertible Kramers-Wannier (KW) as well as the Kennedy-Tasaki (KT) transformation, arriving at the dual models exhibiting spontaneous symmetry breaking (SSB). An alternative model of SSB is written down in each case which, upon the KT transformation, gives rise to new kinds of SPT models. The distinctiveness of the new SPT models from the cluster models is demonstrated by performing the projective symmetry analysis at the interface of the two models. Our investigation lays the groundwork for the broad theme of modulated SPTs in possession of noninvertible symmetries, or NIMSPT. In addition, we employ the topological holography picture to account for the emergence of each of the SPT models considered, by identifying the appropriate two-dimensional (2D) topological model whose boundary physics give rise to the 1D SPT in question. The 2D bulk models that have been identified are two coupled layers of toric codes (for the charge SPT), one layer of anisotropic dipolar toric code (for the dipolar SPT), and two coupled layers of exponentially modulated toric codes (for the exponential SPT). We thus enlarge the horizon of 1D and 2D correspondence through topological holography to the case of MSPT.
 

Coming soon...