Commuting-Projector Boundary Hamiltonians

• Commuting-projector boundary Hamiltonians are local operator algebras derived from truncating bulk Hamiltonians composed of mutually commuting projectors to enforce precise boundary conditions.
• They encode categorical data such as module categories, condensable algebras, and Frobenius structures, which underpin exact solvability in TQFT, SPT, and SET lattice models.
• These models microscopically implement boundary conditions in both anomaly-free and anomalous settings, distinguishing topological phases through symmetry and condensation principles.

Commuting-projector boundary Hamiltonians are boundary-local operator algebras obtained from a bulk Hamiltonian that is a sum of mutually commuting projectors, typically by truncating bulk terms near a boundary and supplementing them with additional boundary projectors that remain local, mutually commuting, and compatible with the bulk constraints. In lattice realizations of TQFTs, SPT phases, SET phases, and anomaly-inflow constructions, they encode the microscopic version of boundary conditions, boundary condensation data, or boundary symmetry actions. Across dimensions, the same structural themes recur: projector idempotence, exact commutativity, Frobenius or module-category data, and, in anomalous settings, boundary operator algebras that are consistent only in the presence of a higher-dimensional bulk (Chen et al., 2021, Williamson et al., 2016).

1. General formulation

A local commuting-projector lattice Hamiltonian is an exactly solvable model of the form

$H=-\sum_i P_i,$

with each $P_i$ a local projector, $P_i^2=P_i$, and $[P_i,P_j]=0$ for all $i,j$. Exact solvability follows because the ground space is the common $+1$ eigenspace of all projectors. On a manifold with boundary $N=\partial M$, a boundary commuting-projector Hamiltonian is obtained by truncating bulk projectors near the boundary and, if needed, adding boundary projectors that are local on $N$, mutually commute among themselves and with the restricted bulk terms, and enforce the appropriate Gauss-law and zero-flux constraints on boundary degrees of freedom. A constructive method emphasized in higher-dimensional anomaly models is the cone construction over $N$: add an auxiliary vertex, cone off the boundary, and restrict the resulting bulk cochains back to $N$ to induce the boundary operators (Chen et al., 2021).

State-sum models supply an equivalent TQFT formulation. In the $P_i$0D UGxBFC setting, the boundary ground-state subspace is the support of the cylinder amplitude $P_i$1 with appropriate boundary projectors inserted. Imposing a boundary condition by a condensable algebra object $P_i$2 amounts diagrammatically to inserting an idempotent built from $P_i$3 and $P_i$4 along boundary loops. In the $P_i$5D state-sum models with fusion-category symmetry, the open-chain ground space and the cylinder idempotent likewise identify the boundary sector selected by the input algebra $P_i$6 (Williamson et al., 2016, Inamura, 2021).

The central significance of this formulation is that a boundary Hamiltonian is not merely an arbitrary edge perturbation. In these constructions it is the microscopic realization of categorical data: module categories, condensable algebras, or defect-junction maps. This is why boundary commutativity is usually proved not by direct brute force alone, but by Frobenius relations, pentagon identities, heptagon identities, Kasteleyn properties, or state-sum Pachner invariance.

2. One-dimensional fixed points and fusion-category-symmetric boundaries

For $P_i$7D gapped phases with non-anomalous fusion-category symmetry, the basic input is a finite-dimensional semisimple Hopf algebra $P_i$8 and an $P_i$9-simple left $P_i^2=P_i$0-comodule algebra $P_i^2=P_i$1. The bulk symmetry category is $P_i^2=P_i$2, while the boundary conditions are objects of the module category $P_i^2=P_i$3. On a circular lattice with local Hilbert spaces $P_i^2=P_i$4, the commuting-projector Hamiltonian is

$P_i^2=P_i$5

Because $P_i^2=P_i$6 carries a $P_i^2=P_i$7-separable symmetric Frobenius algebra structure, the local terms satisfy $P_i^2=P_i$8 and commute for all $P_i^2=P_i$9. On an interval, the natural open-chain choice is

$[P_i,P_j]=0$0

possibly supplemented by boundary idempotents obtained from the state-sum boundary evaluation maps. The fixed-point ground space on an interval is canonically isomorphic, as a vector space, to $[P_i,P_j]=0$1 (Inamura, 2021).

The same construction identifies the boundary symmetry action. If $[P_i,P_j]=0$2, the lattice symmetry operator $[P_i,P_j]=0$3 is defined from the sitewise $[P_i,P_j]=0$4-coaction and the character $[P_i,P_j]=0$5, and it commutes with all local projectors. Restricted to the ground space, $[P_i,P_j]=0$6 reproduces the $[P_i,P_j]=0$7-action on boundary states: $[P_i,P_j]=0$8 Thus the lattice boundary conditions form the $[P_i,P_j]=0$9-module category $i,j$0, exactly matching the classification of $i,j$1D gapped phases with fusion-category symmetry (Inamura, 2021).

For SPT fixed points, $i,j$2 is simple and can be written as

$i,j$3

The interval ground states then take a matrix-product form with edge degrees of freedom in $i,j$4 on the left and $i,j$5 on the right. In the ordinary group case $i,j$6, these edge modes carry the expected projective class $i,j$7. The construction also extends formally to anomalous fusion-category symmetries by replacing $i,j$8 with a semisimple pseudo-unitary connected weak Hopf algebra, although the commutativity proof of the lattice symmetry with the Hamiltonian uses Hopf-specific identities such as $i,j$9, so the anomalous case requires additional analysis (Inamura, 2021).

3. Condensation, module categories, and higher-dimensional gapped boundaries

In $+1$0D lattice models based on unitary $+1$1-crossed braided fusion categories, the boundary Hamiltonian takes the schematic form

$+1$2

Here $+1$3 imposes boundary fusion constraints, $+1$4 enforces compatibility between boundary group variables and defect sectors, $+1$5 inserts and fuses a loop labeled by an allowed boundary object $+1$6, and $+1$7 imposes residual curvature conditions when needed. The allowed labels are determined by a boundary condition specified either by a condensable algebra object $+1$8 or by a module category $+1$9 (Williamson et al., 2016).

The classification of gapped boundaries is expressed in the language of condensable commutative separable Frobenius algebras. A boundary is specified by an algebra object $N=\partial M$0 with multiplication $N=\partial M$1, unit $N=\partial M$2, comultiplication $N=\partial M$3, and counit $N=\partial M$4, satisfying the Frobenius relation

$N=\partial M$5

together with separability $N=\partial M$6, commutativity, and symmetricity. Boundary excitations are simple $N=\partial M$7-modules $N=\partial M$8, and $N=\partial M$9 is the boundary loop-insertion operator restricted to those module channels. The commutativity of boundary plaquettes with boundary vertex terms follows from the Frobenius relations, while commutativity among different boundary plaquettes uses commutativity and separability of $N$0 together with the pentagon and hexagon or heptagon identities of the input category (Williamson et al., 2016).

This categorical structure sharply distinguishes different bulk inputs. For premodular input categories, a gapped commuting-projector boundary can be obtained by condensing an algebra built from the Müger center. The paper gives the example $N$1, and for $N$2 the resulting boundary plaquette is

$N$3

By contrast, for modular input there is no nontrivial condensable Lagrangian algebra; accordingly, strictly local commuting-projector terms cannot fully gap a chiral UMTC boundary. The semion example is the simplest explicit case of this obstruction (Williamson et al., 2016).

In this setting, “boundary Hamiltonian” therefore means more than a truncation prescription: it is an implementation of condensation data. The deconfined boundary excitations are classified by $N$4-modules, and the boundary topological order is determined by the same algebraic object that guarantees commutativity of the microscopic terms.

4. Anomalous boundaries and anomaly inflow

Commuting-projector boundary Hamiltonians are not restricted to anomaly-free boundaries. They also appear as explicit anomalous boundary realizations when a higher-dimensional bulk supplies the inflow. In the $N$5D bosonic beyond-group-cohomology invertible phase with bulk action $N$6, the boundary on $N$7 carries the commuting-projector Hamiltonian

$N$8

These stabilizers mutually commute and enforce boundary Gauss laws and flatness. The boundary supports a pointlike excitation created by $N$9 and a stringlike excitation created by $N$0, with operator algebra

$N$1

The point excitation is fermionic, the loop excitation is also fermionic in the sense diagnosed by orientation-reversal and T-junction processes, and their mutual statistics is $N$2. The same work argues that this topological order cannot be realized as a purely $N$3D commuting-projector model with the same symmetry structure; the boundary Hamiltonian exists precisely because the anomaly is absorbed by the $N$4D bulk (Chen et al., 2021).

A related obstruction appears in the exactly solvable $N$5D bosonic topological insulator protected by $N$6. On a semi-infinite cylinder, the boundary degrees of freedom can be written as dressed Pauli operators $N$7, but the boundary $N$8 action becomes non-onsite: $N$9 This counts boundary domain walls, each carrying fractional charge $N$0. After inserting a $N$1-flux, the twisted time reversal satisfies

$N$2

The model therefore exhibits a Kramers doublet bound to $N$3-flux. The paper concludes that a symmetric, gapped, commuting-projector boundary Hamiltonian does not exist within the boundary-only $N$4 degrees of freedom. What does exist are symmetry-breaking commuting-projector edges, such as

$N$5

and a simple symmetric but non-commuting boundary Hamiltonian that is gapless for $N$6 (Horinouchi, 2020).

For the $N$7D class DIII topological superconductor, the obstruction is formulated in terms of time reversal. A strictly $N$8-invariant, short-range-entangled, commuting-projector physical boundary with a unique ground state is not available in that construction. However, once $N$9 is broken, an internal $P_i$00-domain wall acquires a fixed normal orientation and supports a $P_i$01D commuting-projector wall Hamiltonian

$P_i$02

which is exactly the Tarantino–Fidkowski $P_i$03D fermionic $P_i$04 SPT model (Kobayashi, 2020). This juxtaposition is characteristic: anomaly does not forbid commuting-projector boundaries absolutely; it forbids them in the wrong dimensional or symmetry setting.

5. Fermionic and parafermionic boundary constructions

Fermionic constructions make the boundary dependence on spin structure especially explicit. In the Tarantino–Fidkowski framework for $P_i$05D fermionic SPTs, the spatial surface is equipped with a discrete spin structure encoded by a Kasteleyn orientation. On a disc, the outer face is included in the Kasteleyn condition, and boundary plaquette terms $P_i$06 are defined by the same loop-projector formula as in the bulk, with the outer-face spin $P_i$07 entering the domain-wall projectors. Because the outer face also satisfies the odd-clockwise-edge rule, the same projector-cancellation argument proves exact commutativity for boundary plaquettes and bulk-boundary overlaps. For $P_i$08, the $P_i$09 pattern appears at the boundary: for $P_i$10, symmetric gapped commuting-projector boundaries can be built from disjoint local projectors on $P_i$11-Majorana blocks, while for $P_i$12 protected degeneracy or symmetry breaking remains (Tarantino et al., 2016).

The later “full” commuting-projector Hamiltonian constructions of interacting fermionic SPT phases sharpen this boundary anomaly in operator form. In the $P_i$13D $P_i$14 model, the left-edge symmetry generators satisfy

$P_i$15

In the $P_i$16D $P_i$17 model, the edge realizes projective relations summarized by $P_i$18, $P_i$19, and $P_i$20, with $P_i$21 for the $P_i$22 phase. These relations obstruct a symmetric, gapped, nondegenerate boundary projector Hamiltonian on a single edge, even though the bulk is a full commuting-projector Hamiltonian whose entire spectrum realizes the nontrivial SPT phase (Tantivasadakarn et al., 2018).

Parafermion-decorated models provide a different boundary phenomenon. In the decorated-toric-code model, a fully gapped commuting-projector interface with the parent $P_i$23 fractional quantum Hall fluid is obtained by restricting the bulk projectors to the edge and adding a boundary Zeeman term,

$P_i$24

The commuting-projector structure survives at the edge. By contrast, at the $P_i$25-preserving decorated-domain-wall–$P_i$26 interface, no $P_i$27-invariant Lagrangian subgroup exists after folding, and the interface is symmetry-enforced gapless. The effective boundary theory is the self-dual Hamiltonian

$P_i$28

which flows to the $P_i$29 parafermion CFT with $P_i$30 (Son et al., 2018).

An analogous pattern appears in the commuting-projector model of a $P_i$31D topological insulator. Symmetry-breaking gapped boundaries are obtained either by a boundary Zeeman polarization or by a $P_i$32-breaking pairing rule. The symmetry-preserving boundary instead employs incomplete-plaquette operators $P_i$33 that commute with all bulk $P_i$34 and $P_i$35 terms but do not commute among themselves. The derived $P_i$36D edge Hamiltonian exhibits a gapless helical Luttinger liquid with $P_i$37, consistent with a symmetric anomalous edge rather than a commuting-projector boundary in the strict sense (Son et al., 2019).

6. Constraints, nonchiral behavior, and scope

A fundamental limitation of commuting-projector boundary Hamiltonians follows from the bulk no-go theorem for Hall response. For two-dimensional almost local commuting projector Hamiltonians with finite-dimensional on-site Hilbert spaces and conserved $P_i$38 charge, adiabatic flux insertion pumps no charge in the ground-state sector, and the Hall conductance vanishes: $P_i$39 The proof uses boundary-charge operators $P_i$40 supported near a chosen cut and a weak local topological order condition. One consequence stated explicitly is that commuting-projector boundary Hamiltonians cannot support chiral charge transport and are necessarily nonchiral. The same paper also notes the standard result that commuting Hamiltonians have vanishing thermal Hall conductance, so such boundaries are nonchiral electrically and thermally (Zhang et al., 2021).

This limitation is consistent with the higher-dimensional and parafermionic constructions. In the UGxBFC setting, modular input admits no nontrivial condensable algebra, so strictly local commuting-projector terms cannot fully gap a chiral UMTC boundary (Williamson et al., 2016). In parafermion-decorated models, the construction realizes chiral bosonic non-Abelian order only because the commuting-projector degrees of freedom live inside a parent fractional-quantum-Hall fluid; the chirality is carried by the ambient medium or by an enforced critical interface rather than by a strictly local commuting-projector boundary Hamiltonian itself (Son et al., 2018).

A recurrent source of confusion is therefore the assumption that exact solvability, strict locality, symmetry preservation, and chiral or anomalous boundary behavior can all be imposed simultaneously. The available constructions instead separate these possibilities. Exact commuting-projector boundaries exist for many gapped nonchiral boundaries, for anomalous boundaries supported by higher-dimensional inflow, and for symmetry-breaking terminations. When a boundary is required to remain symmetry-preserving and intrinsically anomalous, the operator algebra typically becomes non-onsite, projective, or explicitly noncommuting, or else the boundary must acquire intrinsic topological order.

In aggregate, commuting-projector boundary Hamiltonians furnish a microscopic boundary calculus for topological phases. In $P_i$41D they are organized by Hopf-algebraic Frobenius data and module categories; in higher-dimensional topological orders they are organized by condensable algebra objects and module categories; in fermionic models they depend on Kasteleyn orientations and discrete spin structures; and in anomalous settings they are constrained by inflow. Their principal strength is exact algebraic control. Their principal limitation is equally sharp: within finite-dimensional, local, commuting-projector frameworks, strictly chiral boundary transport and symmetry-preserving trivialization of anomalous edges are excluded (Inamura, 2021, Williamson et al., 2016, Tarantino et al., 2016, Zhang et al., 2021).