Symmetry Taco: Bilayer Topological Order

• Symmetry Taco is a bilayer topological order construction that synthesizes gapped pure-state, gapless, and mixed-state SPT phases through folded anyon condensation.
• It leverages the SymTFT framework and partial anyon condensation in D(G) to establish equivalences between distinct boundary realizations of SPT orders.
• Positivity constraints on indicator matrices and precise symmetry-fractionalization data (η, ε) underpin its classification and stability.

The symmetry taco is a bilayer topological-order construction in $2+1$ dimensions that extends the symmetry topological field theory (SymTFT), or topological holography, framework from pure-state phases to mixed-state phases in $1+1$ dimensions. In the formulation introduced in "The Symmetry Taco: Equivalences between Gapped, Gapless, and Mixed-State SPTs" (Qi et al., 7 Jul 2025), the central claim is that three ostensibly distinct structures—gapped pure-state SPT phases with symmetry $G\times G$, intrinsically gapless SPTs (igSPTs) with symmetry $G$, and intrinsically average or mixed-state SPTs (iASPTs) with symmetry $G$—can be understood as different boundary realizations of the same bulk construction. The key operation is to begin with a $2+1$d quantum double $D(G)$, partially condense anyons in a finite subregion, and then fold along that subregion into a bilayer topological order whose geometry resembles a folded taco (Qi et al., 7 Jul 2025).

1. SymTFT background and partial anyon condensation

In the SymTFT description of a $1+1$d quantum system with finite on-site symmetry group $G$, the assigned bulk is the $2+1$d quantum double $1+1$0. The standard geometry is a strip with a canonical symmetry boundary on one side and a dynamical boundary on the other. The symmetry boundary is specified by the Lagrangian algebra $1+1$1, where charges are condensed. The dynamical boundary encodes the admissible $1+1$2d phases consistent with the symmetry (Qi et al., 7 Jul 2025).

Anyons in $1+1$3 are irreducible representations of the Drinfeld double of $1+1$4. For Abelian $1+1$5, they may be labeled as

$1+1$6

with topological spin

$1+1$7

and mutual braiding

$1+1$8

In this slab picture, symmetry operators of the $1+1$9d theory are represented by magnetic flux lines parallel to the symmetry boundary, while local charged operators are represented by electric lines stretching between the symmetry and dynamical boundaries. Fully gapped boundaries arise by condensing a Lagrangian subgroup $G\times G$0 consisting of bosons with trivial mutual braiding and cardinality $G\times G$1. For general $G\times G$2, gapped boundaries of $G\times G$3 are classified by pairs $G\times G$4, where $G\times G$5 specifies residual symmetry and $G\times G$6 specifies SPT order under $G\times G$7 (Qi et al., 7 Jul 2025).

The symmetry taco begins from the more general situation of partial rather than maximal condensation. Condensing a condensable subalgebra $G\times G$8 changes the topological order in the condensed region from $G\times G$9 to $G$0. At a boundary, such partial condensation generally yields a gapless boundary because some anyons remain deconfined and generate a higher-form anomaly that forbids a trivial, symmetry-preserving gapped boundary. This is the mechanism by which SymTFT realizes intrinsically gapless SPTs (Qi et al., 7 Jul 2025).

For general $G$1, condensable subalgebras of $G$2 are specified by the data

$G$3

subject to the constraints

$G$4

$G$5

$G$6

For Abelian $G$7, these simplify to

$G$8

The same pair $G$9 is the symmetry-fractionalization data for an igSPT: $G$0 is the subgroup associated with condensed fluxes, $G$1 specifies a projective action of $G$2 in the gapped high-energy sector, and $G$3 encodes how those fractionalized $G$4-charges transform under $G$5 (Qi et al., 7 Jul 2025).

2. Definition of the symmetry taco and its folding constraints

The symmetry taco is obtained by starting with a $G$6d topological order $G$7, condensing a condensable subalgebra $G$8 in a connected subregion, and then folding along a reflection-symmetric axis through that region. Under folding, the bulk becomes

$G$9

where $2+1$0 is the braiding-reversed conjugate theory. The condensed subregion becomes a gapped boundary from $2+1$1 to vacuum, obtained first via an interface to $2+1$2, where $2+1$3, and then via the canonical Lagrangian boundary

$2+1$4

This folded bulk-plus-boundary system is the symmetry taco, or SymTaco (Qi et al., 7 Jul 2025).

In the main applications, $2+1$5. Folding then yields

$2+1$6

This identification is central because it turns data originally formulated for a single $2+1$7-symmetric theory into a bilayer theory with two symmetry copies. The folded construction is not arbitrary: not every gapped boundary of $2+1$8 is realizable in this way. The paper characterizes the allowed folded boundaries by a positivity condition (Qi et al., 7 Jul 2025).

For Abelian topological orders, letting $2+1$9 and labeling anyons as $D(G)$0, one defines an indicator matrix

$D(G)$1

The condensation data must satisfy closure and inversion,

$D(G)$2

as well as the folding conditions

$D(G)$3

$D(G)$4

These are equivalent to the inequality

$D(G)$5

and this is in turn equivalent to the statement that the matrix $D(G)$6 is symmetric and positive semi-definite. The defining structural fact is therefore that folded boundaries are exactly those gapped boundaries whose indicator matrix is positive semi-definite (Qi et al., 7 Jul 2025).

For non-Abelian $D(G)$7, the same role is played by the tunneling matrix $D(G)$8 across the domain wall between $D(G)$9 and $1+1$0. The condensation matrix on the folded boundary is

$1+1$1

which is manifestly positive semi-definite. A plausible implication is that positivity is not merely a technical artifact of the Abelian discussion, but an intrinsic organizing principle of the folded construction.

3. Equivalence between igSPTs and folded gapped SPTs

An intrinsically gapless SPT is a $1+1$2d phase with anomaly-free on-site symmetry $1+1$3 at the microscopic level, but with no symmetric, non-degenerate gapped ground state if the symmetry is preserved. Its topological character is encoded in robust edge modes and nontrivial string order, enforced by an emergent anomaly of the low-energy effective symmetry

$1+1$4

The gapped high-energy sector carries a nontrivial $1+1$5-SPT specified by $1+1$6, while the mixed fractionalization between $1+1$7 and $1+1$8 is encoded in $1+1$9. From $G$0, one derives a $G$1-cocycle $G$2, which is the emergent anomaly of the low-energy symmetry (Qi et al., 7 Jul 2025).

In SymTFT language, an igSPT is a gapless boundary of $G$3 obtained by partially condensing a condensable subalgebra $G$4. The symmetry taco uses exactly the same $G$5, but condenses it in a finite bulk region and then folds the geometry. The result is a gapped folded boundary in

$G$6

and hence a gapped $G$7d SPT with symmetry $G$8, or more precisely with an unbroken subgroup determined by the folding data (Qi et al., 7 Jul 2025).

For the case relevant to igSPTs, the condensable algebra is assumed not to condense charges, corresponding to preservation of the full microscopic $G$9. Then the folded boundary subgroup takes the form

$2+1$0

with elements $2+1$1, where $2+1$2 and $2+1$3, and the semidirect product arises from conjugation $2+1$4. The corresponding $2+1$5-cocycle on the folded boundary is

$2+1$6

This produces a one-to-one map from the condensable data $2+1$7 of the original igSPT to a folded gapped SPT boundary in $2+1$8 (Qi et al., 7 Jul 2025).

The significance of this equivalence is structural rather than merely representational. The same algebraic data control the symmetry fractionalization of the gapless phase and the edge fractionalization of the folded gapped phase. This suggests that the distinction between “intrinsically gapless” and “gapped folded” is geometrical: one is obtained by leaving the partially condensed structure unfolded at a boundary, the other by embedding the same structure into a folded bilayer bulk.

4. Mixed-state formulation: strong symmetry, weak symmetry, and Choi-state SymTFT

For a density matrix $2+1$9, the paper distinguishes two symmetry notions. A weak, or average, symmetry satisfies

$1+1$00

while a strong, or exact, symmetry satisfies

$1+1$01

In general, the total symmetry group need not factorize as a direct product; rather, it may form a group extension

$1+1$02

The Choi-state formalism realizes this symmetry structure on a doubled Hilbert space (Qi et al., 7 Jul 2025).

Writing

$1+1$03

left multiplication by $1+1$04 becomes an action on the $1+1$05 layer and right multiplication by $1+1$06 becomes an action on the $1+1$07 layer. For a weak symmetry,

$1+1$08

so the diagonal subgroup of $1+1$09 acts exactly on the Choi state. For a strong symmetry, separate eigenvalue constraints appear on the two layers. The bilayer bulk $1+1$10 therefore encodes both strong and weak symmetries in a natural way (Qi et al., 7 Jul 2025).

This is the basis for the mixed-state SymTFT proposal: for $1+1$11d gapped mixed-state phases of a finite group $1+1$12, whose Choi states are short-range correlated, the appropriate bulk topological order is precisely the symmetry taco

$1+1$13

with a folded boundary to vacuum. The same positivity condition that characterizes folded boundaries also characterizes physical density matrices. In the Abelian case, Hermiticity implies that if $1+1$14 condenses then $1+1$15 condenses, while positivity implies that if $1+1$16 condenses then $1+1$17 and $1+1$18 condense. The indicator matrix for Choi-state condensates is therefore exactly the positive semi-definite matrix that characterizes folded boundaries (Qi et al., 7 Jul 2025).

A plausible implication is that the symmetry taco is not merely an analogy between pure-state folding and mixed-state vectorization. Rather, the folded boundary condition and the physical positivity of $1+1$19 are mathematically the same organizing constraint in the Abelian formulation.

5. Classification of short-range correlated Choi states and the igSPT–iASPT correspondence

With the symmetry taco as bulk, short-range correlated $1+1$20-symmetric Choi states in $1+1$21d are exactly the gapped boundaries of $1+1$22 that satisfy the folding constraints. For a mixed state with strong symmetry $1+1$23, the full symmetry in Choi space is $1+1$24, and the relevant bulk becomes

$1+1$25

obtained by condensing charges that reduce $1+1$26 to $1+1$27 (Qi et al., 7 Jul 2025).

A gapped mixed-state SPT, or iASPT, is then specified by a subgroup $1+1$28 and a cocycle $1+1$29, subject to the same positivity or folding constraints. These constraints force $1+1$30 to take the same form as in the folded igSPT correspondence: $1+1$31 Hence the classification of short-range correlated, $1+1$32-symmetric Choi states is in one-to-one correspondence with condensable subalgebras $1+1$33, or equivalently with the symmetry-fractionalization data $1+1$34 subject to Davydov’s constraints (Qi et al., 7 Jul 2025).

This is the paper’s principal equivalence: $1+1$35 All three encode the same $1+1$36. The distinction lies in whether the data are read as partial boundary condensation in $1+1$37, as a folded gapped boundary in $1+1$38, or as a positive Choi-state condensate in the bilayer mixed-state bulk. The unification is therefore categorical and geometric rather than merely phenomenological.

The same section of the construction also introduces mixed-state correlators. Using the Choi inner product,

$1+1$39

ordinary expectation values become

$1+1$40

and Rényi-2 correlators take the form

$1+1$41

For an order parameter $1+1$42 charged under the strong symmetry $1+1$43, the paper defines

$1+1$44

$1+1$45

If $1+1$46 exhibits long-range order, the phase has ordinary strong symmetry breaking; if $1+1$47 decays but $1+1$48 has long-range order, the phase exhibits strong-to-weak symmetry breaking (SWSSB). In the SymTaco description, these correlator structures are determined by the anyon-condensation pattern in Choi space (Qi et al., 7 Jul 2025).

6. The $1+1$49 example, local decoherence, and the mixed-state anomaly

The canonical example is the $1+1$50 igSPT constructed by condensing the boson $1+1$51 in $1+1$52. In the classification data,

$1+1$53

for $1+1$54 and $1+1$55. This encodes that the flux $1+1$56 is bound to the charge $1+1$57. The partially condensed region has doubled semion topological order, and the boundary realizes an igSPT with microscopic $1+1$58 symmetry and emergent anomalous $1+1$59 (Qi et al., 7 Jul 2025).

After folding, the bulk becomes $1+1$60, and the folded boundary is a gapped SPT with Lagrangian subgroup

$1+1$61

Using the correspondence

$1+1$62

the paper writes a local commuting-projector Hamiltonian on two $1+1$63 chains whose ground state exhibits the string order parameters

$1+1$64

$1+1$65

$1+1$66

These capture, respectively, spontaneous symmetry breaking and SPT order in the folded phase (Qi et al., 7 Jul 2025).

The same correspondence becomes operational in the mixed-state setting. Starting from an igSPT chain with symmetry $1+1$67, one may decohere the low-energy excitations via local quantum channels whose Kraus operators correspond to the interlayer anyons $1+1$68 to be condensed in Choi space. In Abelian cases, the paper states that explicit channels can be written in terms of Pauli operators using the $1+1$69, $1+1$70 correspondence. The result is a mixed state whose Choi state is the folded SPT, hence an iASPT (Qi et al., 7 Jul 2025).

For the $1+1$71 case, the fixed-point density matrix is

$1+1$72

and it satisfies

$1+1$73

These relations encode both the strong $1+1$74 SPT order and the weak $1+1$75 average symmetry. The corresponding Choi-state condensation matches the folded SPT condensation data exactly (Qi et al., 7 Jul 2025).

The symmetry taco also exposes a new mixed-state anomaly. For a $1+1$76d system with strong $1+1$77 symmetry, the SymTaco is $1+1$78. The paper identifies three relevant Lagrangian subgroups compatible with positivity,

$1+1$79

These correspond, respectively, to trivial symmetric, fully symmetry-broken, and SWSSB phases. After classically gauging the weak $1+1$80 by changing the reference boundary from $1+1$81 to $1+1$82, the positivity constraint implies that no symmetric, short-range correlated mixed state exists whose Choi-state boundary preserves both the strong-modulo-weak symmetries generated by $1+1$83 and $1+1$84. Yet this symmetry is not anomalous in the usual pure-state topological sense. The obstruction is therefore specifically mixed-state and specifically tied to positivity (Qi et al., 7 Jul 2025).

This suggests a broader conceptual shift. In the pure-state setting, anomalies are typically diagnosed by the impossibility of a symmetric gapped boundary compatible with braiding data. In the mixed-state setting described by the symmetry taco, an additional obstruction appears: a symmetric gapped boundary may exist topologically, but fail to correspond to any physical density matrix because positivity excludes it.