Boundary Edge-Mode Construction

• Boundary Edge-Mode Construction is a framework to derive localized states, operators, or fields at boundaries using methods like exact lattice models, analytic continuation, and variational techniques.
• It employs various approaches—from tight-binding systems and interacting chains to wavefunction expansions and gauge theory—to generate robust edge states in different physical contexts.
• The methodology highlights the importance of boundary conditions and interference effects, offering practical insights into topological protection, experimental controls, and operator dynamics.

Boundary edge-mode construction denotes the derivation of states, operators, or fields localized near a boundary, interface, entangling surface, or higher-codimension termination. In the literature represented here, such constructions appear in several mathematically distinct forms: exact destructive-interference wavefunctions on specially organized lattices, analytic continuation of bulk Bloch solutions to exponentially decaying boundary states, operator constructions in which Heisenberg evolution is recast as a one-dimensional tight-binding problem in Krylov space, and variational or phase-space extensions in which new boundary fields restore gauge invariance or encode relational data. The resulting objects include solvable corner, hinge, and surface states; strong or almost strong edge operators in interacting chains; chiral interface channels for Dirac fermions; spinon, magnon, and Rossby edge waves; and gauge-theoretic edge degrees of freedom whose physical status depends sharply on boundary conditions (Kunst et al., 2018, Yates et al., 2020, Carrozza et al., 2021, Shimizu et al., 27 May 2026).

1. Exact lattice and finite-size frameworks

A central construction in tight-binding systems is to enforce destructive interference on auxiliary sublattices so that the boundary state has support only on a distinguished sublattice $A$. In the systematic lattice approach of “boundaries of boundaries,” the lattice is organized into hierarchies $A,B,B',B'',\ldots$, with no direct hopping between $A$ sites, and the exact boundary state takes the form

$$\left|\psi_i(\vec{k}_{||})\right\rangle = \mathcal{N}_i(\vec{k}_{||}) \sum_{\{m^{(s)}\}} \left[ \prod_{s=0}^{D-d-1} \bigl(r_i^{(s)}(\vec{k}_{||})\bigr)^{m^{(s)}} \right] c^\dagger_{A_i,\vec{k}_{||},\{m^{(s)}\}}|0\rangle .$$

The decay parameters $r_i^{(s)}$ are fixed locally by destructive-interference equations, and $|r_i^{(s)}|<1$, $|r_i^{(s)}|>1$, or $|r_i^{(s)}|=1$ determine localization toward one boundary, the opposite boundary, or delocalization in the corresponding direction. This yields exactly solvable states localized to corners, edges, hinges, and surfaces in arbitrary codimension, with the SSH chain, breathing kagome lattice, and breathing pyrochlore lattice serving as explicit examples (Kunst et al., 2018).

For finite topological systems, an alternative construction starts from the exact finite-size eigenproblem rather than a special sublattice hierarchy. Bulk states are written as

$$\psi_{\sigma,k}(x)=A\phi_{\sigma,+k}(x)e^{ikx}+B\phi_{\sigma,-k}(x)e^{-ikx},$$

while edge states are obtained by analytic continuation to exponentially decaying solutions,

$\psi(x)=A\phi_+(x)\alpha^x+B\phi_-(x)\alpha^{-x},$

with $A,B,B',B'',\ldots$0. Open boundaries impose transcendental quantization conditions on $A,B,B',B'',\ldots$1 or $A,B,B',B'',\ldots$2, and the method applies to arbitrary finite lattices, including the incommensurate case. The same framework extends to local impurities, where an impurity next to one edge can drive topological edge states into or out of the bulk continuum and generate a trivial Shockley state bound to the impurity (Duncan et al., 2018).

These exact constructions make boundary localization a local algebraic consequence of the lattice structure. They also separate two notions that are often conflated: exact solvability of a boundary state and its topological protection. The first follows from locality and interference conditions; the second depends on the symmetry and spectral context of the model (Kunst et al., 2018, Duncan et al., 2018).

2. Interacting chains and operator-localized boundary modes

In interacting one-dimensional symmetry-protected topological phases with $A,B,B',B'',\ldots$3 symmetry, boundary edge-mode construction can target operators rather than single-particle wavefunctions. For the generalized spin chain

$A,B,B',B'',\ldots$4

the noninteracting limit $A,B,B',B'',\ldots$5 admits an exact strong mode $A,B,B',B'',\ldots$6 localized at the edge, while $A,B,B',B'',\ldots$7 yields an almost strong mode that only approximately commutes with $A,B,B',B'',\ldots$8 in the thermodynamic limit. The Heisenberg evolution of a boundary operator is mapped by Lanczos recursion to a tight-binding problem in Krylov space,

$A,B,B',B'',\ldots$9

and the resulting effective model is a spatially inhomogeneous Su-Schrieffer-Heeger chain whose hopping amplitude increases away from the boundary and whose dimerization decreases away from the boundary. Short-time dynamics is therefore that of an SSH edge mode, whereas long-time decay proceeds by tunneling through the dimerization barrier into a chaotic Krylov bulk. The lifetime estimate

$A$0

is non-perturbatively long, with $A$1, and competing scattering processes can generate destructive interference that significantly enhances the lifetime (Yates et al., 2020).

At interfaces between phases, the existence problem becomes subtler. In boundary strong zero modes, the interface between a trivial and ordered phase does not guarantee the existence of a strong zero mode, while the interface between two ordered phases can, in certain cases, lead to an exact strong zero mode. In the Kitaev-chain representation, an explicit interface mode is

$A$2

and the spin-chain perturbation theory reveals resonance poles whose appearance is governed by a band-overlap criterion,

$A$3

This criterion is not a conventional topological phase-boundary condition; it reflects whether resonant decay channels connect the boundary operator to bulk excitations (Olund et al., 2023).

A different interacting construction appears in the isotropic XXX chain or equivalent SU(2)-symmetric six-vertex circuit with a boundary interaction defect. There, a quasi-local edge mode is built by a matrix-product ansatz,

$A$4

with $A$5 auxiliary matrices $A$6. For sufficiently strong boundary interaction, the operator is conserved and quasi-localized near the edge, producing non-decaying boundary correlation functions and a nonzero boundary Drude weight; the localization length diverges at a finite critical boundary coupling, beyond which the boundary dynamics becomes ergodic (Prosen, 18 Mar 2026).

In antiferromagnetic magnon systems, the construction becomes explicitly boundary-sensitive in a non-Bloch sense. The linear spin-wave dynamic matrix is non-Hermitian even though the Hamiltonian is Hermitian, the conventional Bloch winding number is trivial, yet finite systems exhibit boundary-localized modes. A generalized Brillouin zone construction based on a complexified momentum $A$7 and a boundary matrix produces a non-Bloch winding number,

$A$8

which correctly predicts when the edge states exist. Additional boundary potentials

$A$9

drive the modes into or out of the bulk spectrum, establishing boundary-controlled topological transitions (Debnath et al., 8 Jun 2026).

3. Wavefunction bases, interface channels, and edge-wave decompositions

In fractional quantum Hall systems, boundary edge modes can be constructed as explicit many-body wavefunctions. For the $$\left|\psi_i(\vec{k}_{||})\right\rangle = \mathcal{N}_i(\vec{k}_{||}) \sum_{\{m^{(s)}\}} \left[ \prod_{s=0}^{D-d-1} \bigl(r_i^{(s)}(\vec{k}_{||})\bigr)^{m^{(s)}} \right] c^\dagger_{A_i,\vec{k}_{||},\{m^{(s)}\}}|0\rangle .$$0 Laughlin phase,

$$\left|\psi_i(\vec{k}_{||})\right\rangle = \mathcal{N}_i(\vec{k}_{||}) \sum_{\{m^{(s)}\}} \left[ \prod_{s=0}^{D-d-1} \bigl(r_i^{(s)}(\vec{k}_{||})\bigr)^{m^{(s)}} \right] c^\dagger_{A_i,\vec{k}_{||},\{m^{(s)}\}}|0\rangle .$$1

edge excitations are generated by multiplying the ground state by a symmetric homogeneous polynomial $$\left|\psi_i(\vec{k}_{||})\right\rangle = \mathcal{N}_i(\vec{k}_{||}) \sum_{\{m^{(s)}\}} \left[ \prod_{s=0}^{D-d-1} \bigl(r_i^{(s)}(\vec{k}_{||})\bigr)^{m^{(s)}} \right] c^\dagger_{A_i,\vec{k}_{||},\{m^{(s)}\}}|0\rangle .$$2,

$$\left|\psi_i(\vec{k}_{||})\right\rangle = \mathcal{N}_i(\vec{k}_{||}) \sum_{\{m^{(s)}\}} \left[ \prod_{s=0}^{D-d-1} \bigl(r_i^{(s)}(\vec{k}_{||})\bigr)^{m^{(s)}} \right] c^\dagger_{A_i,\vec{k}_{||},\{m^{(s)}\}}|0\rangle .$$3

The key structural statement is that every such edge state can be expressed as a linear combination of Jack polynomials with negative parameter and admissible root partitions. In the Laughlin case,

$$\left|\psi_i(\vec{k}_{||})\right\rangle = \mathcal{N}_i(\vec{k}_{||}) \sum_{\{m^{(s)}\}} \left[ \prod_{s=0}^{D-d-1} \bigl(r_i^{(s)}(\vec{k}_{||})\bigr)^{m^{(s)}} \right] c^\dagger_{A_i,\vec{k}_{||},\{m^{(s)}\}}|0\rangle .$$4

This Jack basis allows exact diagonalization within the edge subspace and reliable extraction of edge-mode velocities in the Laughlin and Moore-Read phases, with a generalization to the Read-Rezayi state (Lee et al., 2014).

For single-cone Dirac fermions at a topological-insulator–magnetic-insulator boundary, the construction is an interface matching problem. Starting from

$$\left|\psi_i(\vec{k}_{||})\right\rangle = \mathcal{N}_i(\vec{k}_{||}) \sum_{\{m^{(s)}\}} \left[ \prod_{s=0}^{D-d-1} \bigl(r_i^{(s)}(\vec{k}_{||})\bigr)^{m^{(s)}} \right] c^\dagger_{A_i,\vec{k}_{||},\{m^{(s)}\}}|0\rangle .$$5

one finds that a chiral edge mode appears along the one-dimensional boundary, with velocity

$$\left|\psi_i(\vec{k}_{||})\right\rangle = \mathcal{N}_i(\vec{k}_{||}) \sum_{\{m^{(s)}\}} \left[ \prod_{s=0}^{D-d-1} \bigl(r_i^{(s)}(\vec{k}_{||})\bigr)^{m^{(s)}} \right] c^\dagger_{A_i,\vec{k}_{||},\{m^{(s)}\}}|0\rangle .$$6

provided $$\left|\psi_i(\vec{k}_{||})\right\rangle = \mathcal{N}_i(\vec{k}_{||}) \sum_{\{m^{(s)}\}} \left[ \prod_{s=0}^{D-d-1} \bigl(r_i^{(s)}(\vec{k}_{||})\bigr)^{m^{(s)}} \right] c^\dagger_{A_i,\vec{k}_{||},\{m^{(s)}\}}|0\rangle .$$7 and the momentum $$\left|\psi_i(\vec{k}_{||})\right\rangle = \mathcal{N}_i(\vec{k}_{||}) \sum_{\{m^{(s)}\}} \left[ \prod_{s=0}^{D-d-1} \bigl(r_i^{(s)}(\vec{k}_{||})\bigr)^{m^{(s)}} \right] c^\dagger_{A_i,\vec{k}_{||},\{m^{(s)}\}}|0\rangle .$$8 lies in the finite interval

$$\left|\psi_i(\vec{k}_{||})\right\rangle = \mathcal{N}_i(\vec{k}_{||}) \sum_{\{m^{(s)}\}} \left[ \prod_{s=0}^{D-d-1} \bigl(r_i^{(s)}(\vec{k}_{||})\bigr)^{m^{(s)}} \right] c^\dagger_{A_i,\vec{k}_{||},\{m^{(s)}\}}|0\rangle .$$9

In momentum space the mode is an arc state connecting the Dirac point of the gapless Dirac fermions to the magnetic band gap, and an electric field parallel to the boundary pumps charge via this arc state with current density $r_i^{(s)}$0 (Beenakker, 2024).

In topological Mott insulators, the relevant boundary object is not an electron-like edge pole but a spinon mode encoded in the zeros of the physical Green function. Real-space slave-rotor theory decomposes the electron as

$r_i^{(s)}$1

and the physical Green function becomes a convolution of spinon and rotor Green functions. In the Mott phase the rotor is gapped and the topology is carried by Green-function zeros; a zero edge mode corresponds to a spinon edge mode. At an interface with a conventional topological insulator, the TMI spinon edge mode hybridizes with the spin sector of the TI edge state, leaving a gapless propagating holon and gapped spinon state, i.e. a non-Fermi liquid edge mode (Wagner et al., 2023).

A non-quantum but structurally parallel construction appears in the modified Eady problem with a sloping boundary. There, the streamfunction is decomposed into top and bottom Rossby edge-wave structures,

$r_i^{(s)}$2

with amplitudes $r_i^{(s)}$3 and $r_i^{(s)}$4. The edge-wave phase-shift

$r_i^{(s)}$5

is the mechanistically relevant quantity for instability, not the normal-mode phase-tilt. This edge-wave basis makes the boundary interaction explicit and clarifies how a sloping boundary modifies growth rates and phase locking (Mak et al., 2024).

4. Gauge, gravitational, and entanglement-based edge fields

In diffeomorphism-invariant theories, the construction of boundary edge modes is inseparable from the definition of a local phase space. The extended-phase-space formalism introduces new fields $r_i^{(s)}$6, viewed as a map from a reference space into spacetime, transforming under a diffeomorphism $r_i^{(s)}$7 as

$r_i^{(s)}$8

This restores gauge invariance of the symplectic structure for a subregion with boundary $r_i^{(s)}$9. The extended symplectic form is

$|r_i^{(s)}|<1$0

and the surface-preserving symmetry algebra is universal, comprising $|r_i^{(s)}|<1$1, $|r_i^{(s)}|<1$2 transformations of the normal plane, and, in some cases, normal shearing transformations; with suitable boundary conditions for surface translations, a central extension appears (Speranza, 2017).

For linear Yang-Mills and Chern-Simons theory, a homological construction places the same idea on a systematic algebraic footing. The relevant field content becomes a chain complex with degree-zero fields $|r_i^{(s)}|<1$3 and degree-one gauge parameters $|r_i^{(s)}|<1$4, with differential

$|r_i^{(s)}|<1$5

The boundary scalar $|r_i^{(s)}|<1$6 is the edge mode. In this formulation, edge modes arise from a homotopy pullback implementing boundary conditions “up to gauge,” and the derived critical locus reproduces the extended phase spaces previously introduced for Yang-Mills and Chern-Simons theory (Mathieu et al., 2019).

A related but more explicitly relational viewpoint treats edge modes as dynamical reference frames. Starting from a global variational principle on $|r_i^{(s)}|<1$7, one post-selects the subsector compatible with gauge-invariant boundary conditions on the interface $|r_i^{(s)}|<1$8. The external frame field, constructed for gauge theory as a holonomy-based variable $|r_i^{(s)}|<1$9, leaves an imprint on the subregion dynamics and materializes as a boundary field, i.e. an edge mode. The boundary action can be written in Darboux coordinates as

$|r_i^{(s)}|>1$0

and boundary reorientations divide into proper symmetries, boundary gauge symmetries, and meta-symmetries, depending on whether they preserve the boundary condition and whether their charges vanish (Carrozza et al., 2021).

For antisymmetric $|r_i^{(s)}|>1$1-form gauge fields, the Gauss law implies that the normal component of the field strength on the spherical entangling surface labels superselection sectors. A direct boundary path integral then yields an edge partition function equal to that of co-exact $|r_i^{(s)}|>1$2-forms on the boundary,

$|r_i^{(s)}|>1$3

while the logarithmically divergent edge entanglement entropy agrees with the same determinant representation. In this construction, the would-be gauge parameter becomes a physical boundary field, and the universal term in the entropy is controlled by the Hodge–de Rham spectrum on the entangling surface (Mukherjee, 2023).

5. Boundary conditions, duality, and boundary observables

In Maxwell theory and QED on manifolds with finite spatial boundaries, the physicality of edge modes is controlled by the boundary condition. For the standard Neumann and Dirichlet conditions, large gauge transformations and the corresponding edge-mode shifts are gauge redundancies rather than physical boundary symmetries. Modified boundary conditions change this conclusion. For the modified Neumann condition,

$|r_i^{(s)}|>1$4

the boundary field $|r_i^{(s)}|>1$5 carries a genuine physical symmetry

$|r_i^{(s)}|>1$6

generated by the topological surface charge

$|r_i^{(s)}|>1$7

The electromagnetic S-wall exchanges electric and magnetic descriptions and maps the modified Neumann theory to the dual modified Dirichlet theory, where the magnetic edge mode $|r_i^{(s)}|>1$8 and charge

$|r_i^{(s)}|>1$9

become physical. Singular large gauge transformations are instead interpreted as insertions of Wilson or ’t Hooft loops on the boundary (Shimizu et al., 27 May 2026).

For the Casimir problem with one dynamical-edge-mode plate and one perfect electromagnetic conductor plate, the construction is explicitly BRST-based. DEM boundary conditions introduce new edge fields on the DEM plate, the boundary conditions are lifted into the action using Lagrange multipliers, and the bulk fields are integrated out to obtain a non-local effective boundary theory. Although the edge fields are necessary to restore BRST invariance, the final Casimir force is identical to a PMC-PEMC setup, so that a DEM plate is equivalent to a PMC plate from the point of view of the Casimir effect (Devroe et al., 11 May 2026).

In topologically ordered phases, boundary conditions enter through gapping data rather than gauge fixing. Abelian Chern-Simons theory with a $|r_i^{(s)}|=1$0-matrix supports fully gapped boundaries only when the boundary gapping lattice $|r_i^{(s)}|=1$1 is maximal and null,

$|r_i^{(s)}|=1$2

For a spatial manifold with $|r_i^{(s)}|=1$3 boundaries, the ground-state degeneracy is

$|r_i^{(s)}|=1$4

so boundary degeneracy contains information beyond the bulk fusion algebra. On a cylinder, this distinguishes the $|r_i^{(s)}|=1$5 toric code from the $|r_i^{(s)}|=1$6 double-semion model, even though both have the same torus degeneracy (Wang et al., 2012).

6. Stability, attenuation, and experimental control

Once a boundary mode is constructed, a second problem is to determine whether it remains localized and dynamically relevant. For edge modes propagating along imperfect boundaries in a conformally invariant $|r_i^{(s)}|=1$7-dimensional field-theory model, a conformal map

$|r_i^{(s)}|=1$8

straightens the deformed boundary while transferring the geometric complexity into the bulk kinetic term. The straight-edge solution

$|r_i^{(s)}|=1$9

then serves as the reference mode. Boundary roughness radiates energy into the bulk only for harmonics satisfying

$$\psi_{\sigma,k}(x)=A\phi_{\sigma,+k}(x)e^{ikx}+B\phi_{\sigma,-k}(x)e^{-ikx},$$0

and the damping factor obeys

$$\psi_{\sigma,k}(x)=A\phi_{\sigma,+k}(x)e^{ikx}+B\phi_{\sigma,-k}(x)e^{-ikx},$$1

Attenuation is therefore suppressed as $$\psi_{\sigma,k}(x)=A\phi_{\sigma,+k}(x)e^{ikx}+B\phi_{\sigma,-k}(x)e^{-ikx},$$2 or $$\psi_{\sigma,k}(x)=A\phi_{\sigma,+k}(x)e^{ikx}+B\phi_{\sigma,-k}(x)e^{-ikx},$$3, and short-wavelength modes can be exponentially protected at corners (Adamyan, 2024).

Boundary control is also explicit in magnetic systems. In the non-Bloch antiferromagnetic chain, local Zeeman fields and modified edge anisotropy realize the boundary potential $$\psi_{\sigma,k}(x)=A\phi_{\sigma,+k}(x)e^{ikx}+B\phi_{\sigma,-k}(x)e^{-ikx},$$4 that tunes magnon edge modes into or out of the bulk spectrum (Debnath et al., 8 Jun 2026). In the interacting SPT chain, destructive interference among competing scattering processes produces cusp-like lifetime enhancements for almost strong modes (Yates et al., 2020). In the XXX chain with a boundary defect, the localization length of the quasi-local conserved edge mode diverges at a critical defect strength, marking a transition from persistent edge memory to ergodic boundary dynamics (Prosen, 18 Mar 2026).

Direct observation of chiral boundary transport provides a complementary stability criterion. In $$\psi_{\sigma,k}(x)=A\phi_{\sigma,+k}(x)e^{ikx}+B\phi_{\sigma,-k}(x)e^{-ikx},$$5, laser pulses applied to the surface state under magnetic field produce voltage relaxations whose signs become opposite on the two edges only in the topological insulating phase. The interpretation is based on coupled time-dependent Poisson and Boltzmann equations for a chiral edge distribution, and the sign inversion is attributed to the chirality of the edge state (Sasaki et al., 2018).

Taken together, these constructions show that “boundary edge-mode construction” is not a single method but a family of boundary-localization principles. Depending on the problem, the boundary degree of freedom may be an exactly solvable lattice eigenstate, a quasi-conserved operator, a Jack-polynomial excitation, a chiral interface arc, a spinon zero mode, a co-exact boundary form, or a gauge-compensating frame field. What unifies them is that the boundary is treated as an active structural element of the theory rather than as a passive truncation of the bulk.