Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions

TL;DR

The paper develops a continuum/cochain TQFT framework to compute partition functions and ground state degeneracy across dimensions, linking GSD to extended operators and boundary deconfinement. It introduces a dimensional-decomposition perspective that expresses a $(d+1)$D TQFT as a sum of $(d-1)$D TQFTs labeled by holonomies, enabling explicit GSD calculations for bosonic and fermionic (spin) theories, including Dijkgraaf-Witten twists and fSPT/gauged spin-TQFTs. The work reveals exotic boundary condensation mechanisms, such as composite-operator endings, and demonstrates dualities between symmetry-extension and symmetry-breaking boundary conditions under gauging, with detailed examples in 2+1D and 3+1D. It also develops fermionic reductions and higher-dimensional non-Abelian TQFTs, providing a broad toolkit for distinguishing intrinsic topological orders via GSD, entanglement entropy, and boundary data, with implications for condensed matter, quantum information, and quantum cosmology. Overall, the results offer a rigorous, continuum-field-theoretic path to classify and quantify topological vacua and their tunneling in diverse dimensions. (All expressions are presented in a way that can be directly applied in theoretical and computational settings.)

Abstract

Distinct quantum vacua of topologically ordered states can be tunneled into each other via extended operators. The possible applications include condensed matter and quantum cosmology. We present a straightforward approach to calculate the partition function on various manifolds and ground state degeneracy (GSD), mainly based on continuum/cochain Topological Quantum Field Theories (TQFT), in any dimension. This information can be related to the counting of extended operators of bosonic/fermionic TQFT. On the lattice scale, anyonic particles/strings live at the ends of line/surface operators. Certain systems in different dimensions are related to each other through dimensional reduction schemes, analogous to (de)categorification. Examples include spin TQFTs derived from gauging the interacting fermionic symmetry protected topological states (with fermion parity $\mathbb{Z}_2^f$) of symmetry group $\mathbb{Z}_4\times \mathbb{Z}_2$ and $(\mathbb{Z}_4)^2$ in 3+1D, also $\mathbb{Z}_2$ and $(\mathbb{Z}_2)^2$ in 2+1D. Gauging the last three cases begets non-Abelian spin TQFT/topological order. We consider situations where a TQFT lives on (1) a closed spacetime or (2) a spacetime with boundary, such that the bulk and boundary are fully-gapped and short or long-range entangled (SRE/LRE). Anyonic excitations can be deconfined on the boundary. We introduce new exotic topological interfaces on which neither particle nor string excitations alone condensed, but only fuzzy-composite objects of extended operators can end (e.g. a string-like composite object formed by a set of particles can end on a special 2+1D boundary of 3+1D bulk). We explore the relations between group extension constructions and partially breaking constructions (e.g. 0-form/higher-form/composite breaking) of topological boundaries, after gauging. We comment on the implications of entanglement entropy for some of such LRE systems.

Figures (8)

• Figure 1: We show the quantum energy spectrum as several discrete energy levels in terms of horizontal dashed lines (- - -). The approximate semi-classical energy potential are drawn in terms of the continuous solid black curve. The vertical axis shows the energy value $E$. The horizontal axis illustrates their different quantum numbers, which can be, for example, (1) different eigenvectors spanning different subspaces in the Hilbert space; or (2) different spin/angular/spacetime momenta, etc. This figure shows 3 topological degenerate ground states $|\text{g.s.}_1\rangle$, $|\text{g.s.}_2\rangle$ and $|\text{g.s.}_3\rangle$ with the dark gray horizontal dashed lines (- - -) for their energy levels --- Their energy levels only need to be approximately the same (within the order of $e^{-\# V}$ where $V$ is the system size), but they remain topologically robust. Namely, only via the insertion of the extended operator shown in eqn. \ref{['eq:tunnel-GSD-1']} winding around a non-contractible cycle can the $|\text{g.s.}_2\rangle$ tunnel to the other sectors, even though their energy levels are nearly the same. The energy barrier is proportional to the cost of creating two anyonic excitations at the end of extended operators $W$ in eqn. \ref{['eq:tunnel-GSD-1']}. This energy barrier $\Delta_E$ naively seems to be infinite in TQFT, but it is actually of a finite order $\Delta_E \simeq 4J$ where $J$ is the lattice coupling constant in the UV complete lattice (e.g. in Kitaev's toric code Kitaev2003 or more general twisted quantum double models Wan1211.3695Wan:2014woa). In reality, as an example in 2+1D, the 3 topological degenerate ground states on a $T^2_{\text{space}} \times S^1_{\text{time}}$ can be induced from the filling fraction $\nu=\frac{1}{3}$-Laughlin fractional Quantum Hall states from electrons, or a $U(1)_3$-Chern-Simons theory at the deep IR. Further illustration is shown in Fig. \ref{['fig:Tunnel']}
• Figure 2: Illustration of tunneling between topological vacua, from $|\text{g.s.}_\beta\rangle$ to $|\text{g.s.}_\alpha\rangle$, via an extended operator $W$. In fig.(a), we see the topological vacuum in an original ground state $|\text{g.s.}_\beta\rangle$, where the spatial manifold $M^d$ is shown. On top of $M^d$, there are LRE/LRE bulk/boundary with topologically orders (TQFTs). In fig.(b), after inserting certain extended operator $W$ connecting two boundary components ($\Sigma_1^{d-1}$ and $\Sigma_2^{d-1}$), usually by an adiabatic process, we switch or tunnel to another topological vacuum $|\text{g.s.}_\alpha\rangle$. In the case of a closed manifold, the extended $W$ goes along a non-contractible cycle (representing a nontrivial element of the homology group of $M^d$).
• Figure 3: Relating the dimensional reduction and (de)categorification to measurable physical quantum phenomena in the laboratory. The top part of the subfigure (a) shows the bulk energy spectrum $E$ with energy gap $\Delta_E$, in the large 3+1D size limit. The bottom part shows in grey color a 3D spatial sample on $T^3$ torus with large compact circles in all $x,y,z$ directions. The degenerate zero modes in the energy spectrum are due to the non-trivial topological order (described by a TQFT) of the quantum system. The subfigure (b) shows that the energy spectrum slightly splits due to finite size effect, but its approximate GSD is still topologically robust. The subfigure (c) shows the system on $T^3$ torus in the limit of small circle in the $z$ direction. The energy spectrum forms several sectors, that can be labeled by a quantum number $b$ associated to the holonomy $\oint_{S^1_z} A$ of gauge field $A$ along $z$ (or a background flux through the compact circle) as $b \sim \oint A$. See more detailed explanation in the main text.
• Figure 4: Counting GSD on $T^2_o$, that is $\text{Tr}_{{\cal H}_{T^2_\text{o}}}\,1$. The shaded 2-tori embedded into a $T^3$ represent Poincaré duals to elements of $H^1(M_3,{\mathbb{Z}}_2)\cong{\mathbb{Z}}_2^3$. The red letters A/P denote Anti-periodic/Periodic boundary conditions on the embedded 2-tori.
• Figure 5: (a) The "re-gauging" of a 1+1D (or 2d) abelian finite $K$-gauge theory Vafa:1989ihBhardwaj:2017xp. In general, a group surrounded by a circle means a gauged symmetry, and a group surrounded by a square means a global symmetry. There is a global non-anomalous $K$ group acting on the theory faithfully, which is shown by the $K$ surrounded by a square. When $K$ is gauged, the whole system becomes trivial, meaning the Hilbert space is 1-dimensional on any topology. (b) This case is what we focus on in Sec. \ref{['sec:2+1/1+1DZ2']}. We start from 1+1D $K$ gauge theory coupled with anomalous $G$ symmetry. The anomalous symmetry is thought to be realized as an SPT phase in 2+1D. After gauging the bulk $G$-symmetry, resulting system would be equivalent to a 2+1D $G$-theory with some boundary condition breaking $G$ into a subgroup $G'$ and without coupling with a 1+1D system. There can possibly be a decoupled $K'$ gauge theory on the boundary. In the examples, however, we may neglect $K'$ as absence.