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arXiv:2207.10712 · SciPost Physics Core

Topological holography: Towards a unification of Landau and beyond-Landau physics

Heidar Moradi, Seyed Faroogh Moosavian, Apoorv Tiwari

Published Oct 16, 2023• 88 citations
arXivHTMLPDF

Abstract

We outline a holographic framework that attempts to unify Landau and beyond-Landau paradigms of quantum phases and phase transitions. Leveraging a modern understanding of symmetries as topological defects/operators, the framework uses a topological order to organize the space of quantum systems with a global symmetry in one lower dimension. The global symmetry naturally serves as an input for the topological order. In particular, we holographically construct a String Operator Algebra (SOA) which is the building block of symmetric quantum systems with a given symmetry G in one lower dimension. This exposes a vast web of dualities which act on the space of G-symmetric quantum systems. The SOA facilitates the classification of gapped phases as well as their corresponding order parameters and fundamental excitations, while dualities help to navigate and predict various corners of phase diagrams and analytically compute universality classes of phase transitions. A novelty of the approach is that it treats conventional Landau and unconventional topological phase transitions on an equal footing, thereby providing a holographic unification of these seemingly-disparate domains of understanding. We uncover a new feature of gapped phases and their multi-critical points, which we dub fusion structure, that encodes information about which phases and transitions can be dual to each other. Furthermore, we discover that self-dual systems typically posses emergent non-invertible, i.e., beyond group-like symmetries. We apply these ideas to 1+1d 1 + 1 d quantum spin chains with finite Abelian group symmetry, using topologically-ordered systems in 2+1d 2 + 1 d . We predict the phase diagrams of various concrete spin models, and analytically compute the full conformal spectra of non-trivial quantum phase transitions, which we then verify numerically.

DOI
10.21468/scipostphyscore.6.4.066

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We outline a holographic framework that attempts to unify Landau and beyond-Landau paradigms of quantum phases and phase transitions. Leveraging a modern understanding of symmetries as topological defects/operators, the framework uses a topological order to organize the space of quantum systems with a global symmetry in one lower dimension. The global symmetry naturally serves as an input for the topological order. In particular, we holographically construct a String Operator Algebra (SOA) which is the building block of symmetric quantum systems with a given symmetry G in one lower dimension. This exposes a vast web of dualities which act on the space of G-symmetric quantum systems. The SOA facilitates the classification of gapped phases as well as their corresponding order parameters and fundamental excitations, while dualities help to navigate and predict various corners of phase diagrams and analytically compute universality classes of phase transitions. A novelty of the approach is that it treats conventional Landau and unconventional topological phase transitions on an equal footing, thereby providing a holographic unification of these seemingly-disparate domains of understanding. We uncover a new feature of gapped phases and their multi-critical points, which we dub fusion structure, that encodes information about which phases and transitions can be dual to each other. Furthermore, we discover that self-dual systems typically posses emergent non-invertible, i.e., beyond group-like symmetries. We apply these ideas to 1+1d 1 + 1 d quantum spin chains with finite Abelian group symmetry, using topologically-ordered systems in 2+1d 2 + 1 d . We predict the phase diagrams of various concrete spin models, and analytically compute the full conformal spectra of non-trivial quantum phase transitions, which we then verify numerically.
The discovery of generalized symmetries has led to powerful new insights into quantum matter. They have been used to classify new families of quantum phases, place constraints on phases realizable in a given physical system, and conceptually unify seemingly disparate phenomena. In many ways, they prove just as powerful as traditional symmetries at organizing and constraining the theories that describe quantum matter. In this thesis, we attempt a unification of such constraints by developing a holographic correspondence between (generalized) symmetries and topological orders, called the Sym/TO correspondence. For any (finite internal) symmetry of a quantum system in d (spatial) dimensions, we associate with it a unique topological order in d + 1 dimensions, called its Symmetry Topological Order (SymTO). We devise an operator algebraic recipe to compute the SymTO data for any lattice spin model, demonstrating it in a number of examples. We then use the SymTO to classify possible quantum phases allowed by the symmetry—we call this a generalized Landau paradigm. Besides classifying phases, we also identify constraints on the phase transitions between them using a SymTO-resolved modular bootstrap. We test this framework in a quantum spin chain with non-invertible symmetries. We discover a new Kramers-Wannier-like duality and a rich phase diagram including a noninvertible symmetry-enriched incommensurate phase. The translation symmetry of the spin chain has a nontrivial interplay with the lattice Kramers-Wannier duality, which matches the anomaly of the corresponding non-invertible symmetry in the low-energy effective field theory. Finally, we explore such unusual anomaly-matching mechanisms in more detail in the context of the chiral anomaly of a single massless Dirac fermion, demonstrating a novel lattice realization of chiral symmetries and their anomaly.
Topological holography is a holographic principle that describes the generalized global symmetry of a local quantum system in terms of a topological order in one higher dimension. This framework separates the topological data from the local dynamics of a theory and provides a unified description of the symmetry and duality in gapped and gapless phases of matter. In this work, we develop the topological holographic picture for (1+1) d d quantum phases, including both gapped phases as well as a wide range of quantum critical points, including phase transitions between symmetry protected topological (SPT) phases, symmetry enriched quantum critical points, deconfined quantum critical points, and intrinsically gapless SPT phases. Topological holography puts a strong constraint on the emergent symmetry and the anomaly for these critical theories. We show how the partition functions of these critical points can be obtained from dualizing (orbifolding) more familiar critical theories. The topological responses of the defect operators are also discussed in this framework. We further develop a topological holographic picture for conformal boundary states of (1+1) d d rational conformal field theories. This framework provides a simple physical picture to understand conformal boundary states and also uncovers the nature of the gapped phases corresponding to the boundary states.
Two global symmetries are holoequivalent if their algebras of local symmetric operators are isomorphic. A holoequivalent class of global symmetries is described by a topological order (TO) in one higher dimension (called symmetry TO), which leads to a symmetry/topological-order (Symm/TO) correspondence. We establish that (1) for systems with a symmetry described by symmetry TO M\mathcal{M}M, their gapped and gapless states are classified by condensable algebras A\mathcal{A}A, formed by elementary excitations in M\mathcal{M}M with trivial self-/mutual statistics. These so-called A\mathcal{A}A states can describe symmetry breaking orders, symmetry protected topological orders, symmetry enriched topological orders, gapless critical points, etc. in a unified way. (2) The local low-energy properties of an A\mathcal{A}A state can be calculated from its reduced symmetry TO M/A{\mathcal{M}}_{/\mathcal{A}}M/A​, using holographic modular bootstrap (holoMB) which takes M/A{\mathcal{M}}_{/\mathcal{A}}M/A​ as an input. Here M/A{\mathcal{M}}_{/\mathcal{A}}M/A​ is obtained from M\mathcal{M}M by condensing excitations in A\mathcal{A}A. Notably, an A\mathcal{A}A state must be gapless if M/A{\mathcal{M}}_{/\mathcal{A}}M/A​ is nontrivial. This provides a unified understanding of the emergence and symmetry protection of gaplessness that applies to symmetries that are anomalous, higher-form, and/or noninvertible. (3) The relations between condensable algebras constrain the structure of the global phase diagram. We find that, for $1+1\mathrm{D} {\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{\ensuremath{'}}$ symmetry with mixed anomaly, there is a stable continuous transition (deconfined quantum critical point) between the Z2{\mathbb{Z}}_{2}Z2​-breaking-${\mathbb{Z}}_{2}^{\ensuremath{'}}$-symmetric phase and the Z2{\mathbb{Z}}_{2}Z2​-symmetric-${\mathbb{Z}}_{2}^{\ensuremath{'}}$-breaking phase. The critical point is the same as a Z4{\mathbb{Z}}_{4}Z4​ symmetry breaking critical point. (4) 1+1D1+1\mathrm{D}1+1D bosonic systems with S3{S}_{3}S3​ symmetry have four gapped phases with unbroken symmetries S3,Z3,Z2{S}_{3}, {\mathbb{Z}}_{3}, {\mathbb{Z}}_{2}S3​,Z3​,Z2​, and Z1{\mathbb{Z}}_{1}Z1​. We find a duality between two transitions ${S}_{3}\ensuremath{\leftrightarrow}{\mathbb{Z}}_{1}$ and ${\mathbb{Z}}_{3}\ensuremath{\leftrightarrow}{\mathbb{Z}}_{2}$: they are either both first order or both (stably) continuous, and in the latter case, they are described by the same conformal field theory (CFT). (5) The gapped and gapless states for 1+1D1+1\mathrm{D}1+1D bosonic systems with anomalous S3{S}_{3}S3​ symmetries are obtained as well. For example, anomalous S3(1){S}_{3}^{(1)}S3(1)​ and S3(2){S}_{3}^{(2)}S3(2)​ symmetries can have symmetry protected chiral gapless states with only symmetric irrelevant and marginal operators.
A recurring pattern in recent condensed matter theory is the inadequacy of the Landau paradigm—spontaneous breaking of invertible symmetry characterized by a local order parameter—to describe a growing class of quantum phase transitions. This synthesis examines a cluster of eight recent preprints (primarily from cond-mat.str-el, with supporting results from cond-mat.mes-hall and cond-mat.dis-nn) that collectively develop the microscopic underpinnings of beyond-Landau transitions. The central thesis, offered as a **heuristic reading rather than a derivation**, is the following structural pattern: when the symmetry protecting or separating phases is non-invertible—meaning defect fusion does not close in a group—the conventional Landau machinery (local order parameter, symmetry-breaking pattern, Ginzburg-Landau effective field theory) must be replaced by categorical data encoding fusion, F-symbols, and tube-algebra modules. This replacement is not merely formal: it has concrete microscopic consequences for which transitions are allowed, what order parameters signal them, and whether a conventional hidden symmetry-breaking description is even possible after gauging. The corpus sources span deconfined quantum critical points (DQCPs), twin phases, non-invertible symmetry-enriched topological orders (NI-SETOs), Rydberg-chain phase diagrams, topological edge memory under quenches, and the algebraic structure of string order parameters. We identify a falsification path for the central claim: if a numerical (DMRG/tensor-network) study of any of the proposed Rep(H₈)- or Rep(D₈)-type lattice models finds a conventional Landau description after an exhaustive gauging search, the categorical necessity argument collapses. The Limitations section acknowledges abstract-only reading and the absence of full proofs from several sources. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.27166, 2605.27193, 2605.27672, 2605.28688, 2605.28794, 2605.29745, 2605.31601, 2605.31602, 2606.05856, 2606.06343, 2606.06466, 2606.08558, 2606.11303, 2606.12089, 2606.18529, 2606.19437, 2606.20403, 2606.20522
Despite great success we have achieved in the past two decades on systematically understanding gapped topological phases, a firm foundation for understanding gapless topological phases is still lacking. In this dissertation I develop a general framework to describe, classify and construct gapped and gapless phases with or without symmetry. I then apply this framework to construct new gapless phases, and classify some classes of them completely. The first chapter is a non-technical summary of the subjects, ideas and results of the rest of the dissertation. In the first chapter I introduce the notion of gapless topological phases, and briefly discuss their general structures. We also discuss how topological holography can be useful for classifying and constructing gapless phases with generalized symmetries. In chapter two I develop a framework for understanding general quantum phases and phase transitions with generalized symmetries. After reviewing the established theory of gapped phases, we introduce two key notions: quantum order and symmetry-enriched quantum order. These concepts are the correct generalized of topological order and symmetry-enriched topological orders. I show that these new notions subsume conventional phase transitions, gapless spin liquids, and gapless symmetry-protected topological phases. For a class of symmetry-enriched quantum orders with finite many topological defects, we present a general classification using higher tensor category theory. The rest of the dissertation provides more detailed studies of specific gapless topological phases, including classification and lattice model construction of gapless symmetry-protected topological phases and the connection to the theory of topological holography.
Recently, a generalized Landau paradigm has been proposed, in which a broad class of "unconventional" phase transitions lying beyond the original symmetry-breaking framework of Landau are understood as the symmetry-breaking transitions of generalized symmetries. In particular, in (1+1)D, such "topological" or "deconfined" transitions can be mapped to the symmetry-breaking transitions of non-invertible symmetries. How to extract the universal dynamical properties of the transitions from this understanding? In this paper, we follow the Ginzburg-Landau philosophy and develop a field-theoretic description of such critical points in terms of the fluctuations of local order parameters, now of non-invertible symmetries. The Symmetry Topological Field Theory formalism plays a central role in our analysis, allowing us to identify the algebra of local order parameters from the Lagrangian algebra of a (2+1)D topological order. We illustrate the key ideas and the full steps of this generalized Ginzburg-Landau procedure using the transitions between gapped phases with a simple non-invertible symmetry --- the Rep(S3S_3S3​) symmetry.