Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics

TL;DR

This paper advances a bulk–boundary framework, termed topological holography, to unify Landau-type symmetry breaking with beyond-Landau (topological) phases in quantum matter. It constructs a boundary String Operator Algebra (SOA) from bulk 1-form symmetries of a 2+1D topological order and shows that all G-symmetric 1+1D theories can be organized and explored via this holographic gadget. The bulk supplies dualities through 0-form anyonic symmetries and domain-wall defects, enabling powerful constraints on phase diagrams, gapped-phase classifications via Lagrangian subgroups, and exact conformal spectra for nontrivial critical points by mapping twisted partition functions across dual theories. The framework is illustrated through extensive 1+1D examples with finite Abelian groups (Z2, ZN, Z2×Z2, Z3×Z3, Z2×Z4, Z2×Z2×Z2), where dualities organize fixed-point Hamiltonians, predict phase structure, and are numerically validated with DMRG and exact diagonalization. Beyond providing a unifying lens, the work reveals emergent non-invertible symmetries at self-dual points and paves the way for extensions to non-Abelian, fermionic, and higher-dimensional systems, potentially enriching the classification of quantum phases and transitions.

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 $1+1d$ quantum spin chains with finite Abelian group symmetry, using topologically-ordered systems in $2+1d$. 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.

Figures (38)

• Figure 1: a) Symmetry operators are topological $\mathcal{U}_{\mathsf{g}}[X_{d}]=\mathcal{U}_{\mathsf{g}}[X_{d}']$. This is due to the conserved current $d{\star}j=0$ implying $\int_{Y_{d+1}}d{\star}j = \int_{X_d }{\star}j-\int_{X'_d}{\star}j=0$, where $Y_{d+1}$ is a manifold bounded by $X_d$ and $X_{d'}$. b) When both $0$-form (here a surface) and $1$-form (here a line) symmetries are present, the $0$-form symmetry can act on the $1$-form symmetry to form a higher mathematical structure called a $2$-group.
• Figure 2: (a) A ($0$-form) global symmetry is associated with a $d$-dimensional surface $X_d$ in a $d+1$-dimensional spacetime. A $0$-dimensional (local) operator will transform under some representation $\alpha$ when crossing this hypersurface. If $X_d$ wraps around the point $x$, then by shrinking the surface $\mathcal{U}_{\mathsf g}[X_d]\mathcal{O}_{\alpha}(x) = \mathsf R_\alpha(\mathsf g)O_{\alpha}(x)$. (b) If $X_d$ is a time-slice associated with a Hilbert space then we have $\mathcal{U}_{\mathsf g}[X_d]\mathcal{O}_{\alpha}(x) = \mathsf R_\alpha(\mathsf g)O_{\alpha}(x)\mathcal{U}_{\mathsf g}[X_d]$, which is the standard transformation law in a Hilbert space.
• Figure 3: A gauge field corresponding to a finite Abelian group can be defined on a triangulated spacetime and is equivalent to collection of symmetry defects. Gauge transformations implement topological re-configurations of the symmetry defects. The figure illustrates a $\mathbb{Z}_2$ gauge field $A$. The edges (1-simplices) in green and black have $A_{ij}=1$ and $0$ respectively. Equivalently, this gauge field configuration can be described by a single symmetry defect denoted in red. A gauge transformation $\phi_i=1$ on a single site (in blue) and 0 everywhere else transforms the gauge field and correspondingly topologically deforms the symmetry defect.
• Figure 4: Illustration of the 2+1 dimensional spacetime of a topological gauge theory. The orange surface is a time-slice associated with a Hilbert space. (a) The theory has topological line operators labeled by $d=(\mathsf g,\alpha)$. The set of these operators are $\mathcal{A}$, is the group of $1$-form symmetries. In a topological phase, open line operators create anyonic excitations at the end-points of the line. Under time evolution, these end-point evolve into world-lines (dotted lines). (b) Surface operators associated to global ($0$-form) symmetries of the theory $\sigma\in\mathcal{G}[\mathsf G]$. When the surface corresponding to the symmetry $\mathscr D_{\sigma}$ is put on the time-slice it acts on the Hilbert space as a conventional symmetry operator, transforming the charges $(\mathsf g,\alpha)$ to $\sigma\cdot(\mathsf g,\alpha)\in\mathcal{A}$ with isomorphic operator algebras.
• Figure 5: Regularization of the boundary such that magnetic lines $S_{(\mathsf g,0)}$ can only end on odd sites while electric lines $S_{(0,\alpha)}$ can only end on even sites. Alternatively, this can be thought of as the boundary of a topological gauge theory with alternating topological (gapped) boundary conditions corresponding to the condensation of magnetic and electric lines. With such boundary conditions, electric and magnetic lines are allowed to end on certain segments of the boundary and remain gauge-invariant. These new non-contractible line operators correspond to relative homology cycles and become our boundary operators.