Tensor Network Discretization of Topological Field Theories

• Tensor network discretization is a method where lattice-based tensors encode the gauge and symmetry structures of continuum topological field theories.
• This framework enables the construction of nonlocal operators, such as string and brane operators, that determine ground state degeneracy and anyonic statistics.
• It provides a computational bridge linking discrete path integrals with lattice models, facilitating simulation and analysis of topologically ordered quantum matter.

A tensor network discretization of topological field theories is a formalism in which many-body quantum states or path integrals of topological phases are encoded as contractions of local tensors arranged on a lattice. This formalism provides both a descriptive language for topological order in quantum matter and a practical toolset for computing physical, informational, and statistical properties of the corresponding quantum states or partition functions. Central to this approach are the structural properties of gauge and symmetry transformations in the underlying tensor network, the emergence of nonlocal operators, and their explicit connection to discretizations of continuum topological field theory concepts.

1. Local Gauge and Symmetry Structures

Tensor network states (TNS) exhibit structural redundancy: different sets of tensors can generate the same many-body wave function, a property formalized via gauge transformations acting nontrivially on the “inner” indices. For a TNS based on local tensors $T^{m}_{\textbf{indices}}$, the set of gauge transformations is specified by invertible matrices $A_{v_i v_j}$ and scaling factors $\Lambda_{v_i}$, which act on the virtual bond indices. When these transformations are restricted to a $d$-dimensional subregion (a "d-brane"), they define a $d$-dimensional gauge transformation (d-GT). The group of all such transformations leaving the tensors invariant up to scaling is termed the "full invariant gauge group" (d-fIGG), and the $d$-dimensional invariant gauge group is the quotient

$$\text{d-IGG} \equiv \text{d-fIGG}/\text{(d–1)-fIGG}$$

For the physical symmetry, transformations on the local physical legs $m$ that change the tensor, together with a compensating gauge operation ensuring invariance of the full wave function, define the $d$-dimensional local physical transformation (d-lPT). The subset of d-lPTs that leave the total state invariant forms the "d-dimensional symmetry group" (d-SG), again extracted as a quotient.

For TNS constructed in $d_{\text{space}}$ spatial dimensions, the highest-dimensional IGG corresponds to the global gauge symmetry. Lower-dimensional structures (e.g., 1-IGG, 2-IGG) are crucial for the construction of string and brane operators that implement topological observables.

2. Construction of Nonlocal Operators: Strings and Branes

The interplay between gauge and symmetry structures enables concrete construction of nonlocal operators, such as string operators (1D curves) and higher-dimensional brane operators, directly in the discrete tensor network context. These arise by restricting gauge or symmetry-gauge transformations to a finite region of the network (such as a path or membrane), the boundary of which translates to an operator insertion on the physical degrees of freedom.

For instance, in the $Z_2$ toric code–type example, magnetic and electric string operators correspond, respectively, to

$$Z(C^*) = \prod_{i\in C^*} \sigma^z_i;\qquad X(C) = \prod_{i\in C} \sigma^x_i$$

where $C$ and $C^*$ are, respectively, curves in the primal and dual lattice. These operators emerge naturally by examining how gauge transformations restricted to a region modify the TNS, and their structure is dictated by the IGG and SG.

These operators play the dual role of generating degenerate ground states when applied along non-contractible cycles (as on a torus), and creating quasiparticle excitations when open-ended (by locally violating certain constraints).

3. Topological Properties: Ground State Degeneracy and Quasiparticle Statistics

The tensor network formalism provides explicit tools to extract topological invariants of the underlying field theory. The existence and algebra of string and brane operators, determined by the IGG and SG, encode both the ground state degeneracy and the statistics of emergent excitations:

• On a torus, the number of independent non-contractible string operators (modulo local constraints) directly yields the ground state degeneracy; e.g., four for the $Z_2$ toric code ground state.
• Operator algebra, such as the commutation relations between $X(C)$ and $Z(C^*)$, encapsulates the braiding or mutual statistics of the corresponding anyons, matching the predictions for fractionalization in topologically ordered phases.

Examining how open string or brane operators act on the TNS identifies the physical manifestations of topological excitations and their dynamical properties.

4. Discretization of Topological Field Theories

Tensor network constructions realize a faithful discretization of topological field theories (TFT) at the lattice level. The wave function amplitudes generated by contracting a network of local tensors play the role of a state-sum or discrete path integral (e.g., Dijkgraaf–Witten TQFT, Turaev–Viro–Barrett–Westbury models).

Key aspects of this discretization include:

• The assignment of local gauge and symmetry (via IGG and SG) corresponds to the gauge structure of the original continuum field theory.
• Nonlocal operators constructed via the network mirror the Wilson loop, 't Hooft loop, and higher-form operators of TFT.
• Physical predictions, such as topological entanglement entropy, modular matrices, and response to defects, become quantitative calculations involving the tensors' structure and symmetries.

Explicit formulas clarifying this connection include

$$\text{d-IGG} \equiv \text{d-fIGG}/\text{(d–1)-fIGG}$$

and

$$X(C) = \prod_{i\in C} \sigma^x_i\,,\quad Z(C^*) = \prod_{i\in C^*} \sigma^z_i$$

demonstrating the extraction of TFT operators from local tensor gauge structures.

5. Exemplary Models and Applications

A paradigmatic construction is the tensor network for the $Z_2$ topological phase:

• The lattice Hamiltonian is realized as

$H = U \sum_{v} (1-Q_v) + g \sum_p (1-B_p)\,,$

where $Q_v$ and $B_p$ enforce local constraints.

• Tensors $T$, $T'$, and a link tensor $g^m$ (with $g^{m=1}_{00}=g^{m=-1}_{11}=1$) are specified to ensure that the local degrees of freedom obey closed-string constraints, and the IGG is explicitly $Z_2$.
• All physical properties (degeneracy, statistics, string operator structure) follow directly from the tensor symmetries.

This approach underlies the broader G-injective tensor network formalism, extending to non-Abelian quantum double models and more exotic symmetry-protected and symmetry-enriched topological phases. It provides a powerful framework for:

• Numerical simulation of topologically ordered matter.
• Construction and classification of new models for quantum error correction.
• Analysis of the correspondence between discrete lattice models and continuum TFT.

6. Significance and Future Directions

The systematic identification of local gauge and symmetry structures in tensor networks offers a concrete and computationally tractable discretization of TFT, informing both theoretical classification and practical implementation of topological quantum matter. Major implications include:

• Enabling scalable algorithms for simulating and characterizing highly entangled phases, inaccessible to conventional mean-field or perturbative approaches.
• Clarifying the relationship between local symmetries of lattice models and global invariants of field theories.
• Providing blueprints for the realization and detection of topological features (defects, braiding, boundary modes) in both theoretical and experimental settings, with impact on quantum computation and condensed matter physics.

The extension to models with more general symmetry input—G-injective, weak Hopf algebra, and higher category tensor networks—remains a frontier, as does the exploration of connections to higher-form symmetries and their physical realization in 3+1D and beyond.

Table: Key Structures in Tensor Network Discretization

Structure	Tensor Network Concept	TFT Correspondence
d-IGG	Gauge group on d-branes	Local gauge transformations
d-SG	Symmetry group on d-branes	Projective/physical symmetries
String operator	Gauge/symmetry variation on curve	Wilson/'t Hooft loops
Brane operator	Gauge/symmetry variation on d-surface	Higher-form observables
Ground state degeneracy	Action of non-contractible operators	Topological invariants
Quasiparticle statistics	Operator algebra	Anyon fusion and braiding data

This correspondence illustrates how tensor network methods unify lattice and continuum topological physics at the level of both computational practice and conceptual structure, serving as a central bridge in the exploration and engineering of topologically protected quantum states (Swingle et al., 2010).