Boundary-Controlled Topological Transitions

• Boundary-controlled topological transitions are phase changes driven by modifications at system boundaries, altering topological order and edge modes.
• They span diverse platforms including toric codes, non-Hermitian lattices, and grain boundaries, each employing unique diagnostics like entanglement measures and winding numbers.
• Experimental realizations from coupled waveguides to granular chains demonstrate precise control over phase behavior and interfacial dynamics.

Searching arXiv for recent and directly relevant papers on boundary-controlled topological transitions. Search 1: boundary-controlled topological transitions site:arxiv or relevant titles. Boundary-controlled topological transitions are phase changes in which the operative control variable is localized at a physical boundary, bipartition interface, grain boundary, or boundary condition, rather than introduced as a homogeneous bulk perturbation. Across recent work, the term encompasses changes in topological order and boundary criticality in toric-code and symmetry-protected topological systems, non-Bloch and generalized-boundary transitions in non-Hermitian lattices, boundary-mode creation and annihilation in open tight-binding and magnonic chains, and interfacial structural transformations in polycrystals and thin films (Jamadagni et al., 2020, Lu, 2024, Prembabu et al., 2022, Verma et al., 2023, Bissonnette et al., 2023, Devulapalli et al., 2024, Chen et al., 2020).

1. Scope and defining features

Boundary control appears in several technically distinct forms. In lattice topological matter it can mean changing rough versus smooth boundaries in the toric code, tuning a single boundary coupling in a critical SPT chain, attaching semi-infinite leads to an SSH chain, or varying a boundary potential in a non-Hermitian magnon problem. In mixed-state settings it can mean applying local Pauli noise only on qubits that straddle a bipartition boundary. In materials science it can mean altering the chemistry or topology of a grain boundary through solute segregation, disconnection unbinding, wall bias, or local junction reconnection (Jamadagni et al., 2020, Lu, 2024, Prembabu et al., 2022, Bissonnette et al., 2023, Debnath et al., 8 Jun 2026, Devulapalli et al., 2024, Hennessy et al., 2014).

The diagnostics are correspondingly diverse. The literature uses ground-state degeneracy, topological entanglement entropy, topological entanglement negativity, Zak phase, skyrmion number, non-Bloch winding numbers, open-loop expectation values, corner-state counts, excess solute, and interface-localized spectral modes. A compact summary is given below.

System class	Boundary control	Primary diagnostic
Toric code and topological order	rough/smooth boundaries; boundary decoherence	GSD, open-loop expectation value, $E_{\rm topo}$
SPT and boundary criticality	single boundary coupling $b$; self-dual boundary Hamiltonian	boundary order parameters, CFT data
Open and non-Hermitian chains	leads, generalized boundary conditions, boundary potential $\epsilon$	zero-mode counting, GBZ winding, edge spectra
Grain boundaries and interfaces	segregation, disconnections, wall bias, junction reconnection	hysteresis, $\Gamma$, KT flow, energy dissipation rate
Wave and metamaterial platforms	smooth boundary interpolation, contact-angle tuning	localized subgap or interface modes

A recurring structural feature is that the boundary is not merely a passive termination. It enters the effective theory as a control field, a condensate selector, a localized perturbation, or a distinct thermodynamic subsystem. This suggests that “topology” in these problems is often encoded jointly by bulk data and by admissible boundary processes, rather than by bulk band structure alone.

2. Boundary conditions, decoherence, and categorical criticality in quantum topological matter

In the toric code, opening the lattice requires a choice of which anyon type condenses at the boundary. Rough boundaries are $e$-condensing and truncate $A_v$ to a three-spin operator; smooth boundaries are $m$-condensing and truncate $B_p$ to three spins. On a closed surface of genus $g$, the ground-state degeneracy is $4^g$. On an open surface with $b$0 boundary components, the degeneracy is $b$1, giving $b$2 for cylinders with two rough or two smooth boundaries and $b$3 for a cylinder with mixed boundaries. The universal topological entanglement entropy remains $b$4 across these boundary-controlled transitions, while the open-loop operator $b$5 distinguishes whether the endpoint $b$6-anyons condense, with $b$7 on rough boundaries and $b$8 when $b$9 is non-condensing (Jamadagni et al., 2020).

Boundary decoherence yields a different class of transition. For toric codes in $\epsilon$0, local Pauli-$\epsilon$1 or Pauli-$\epsilon$2 noise is applied only on the qubits that straddle the $\epsilon$3-dimensional bipartition boundary, and mixed-state long-range entanglement is measured by

$\epsilon$4

The corresponding topological entanglement negativity is the subleading constant in

$\epsilon$5

A key exact result is the mapping of the negativity spectrum to a $\epsilon$6-dimensional cluster-state SPT wavefunction under symmetry-preserving boundary perturbation, with $\epsilon$7. The mapped statistical models imply no disentangling transition for the $\epsilon$8 toric code with $\epsilon$9-noise and for the $\Gamma$0 toric code with $\Gamma$1-noise, a second-order Ising-universality disentangling transition for the $\Gamma$2 toric code with $\Gamma$3-noise at $\Gamma$4, and a finite-$\Gamma$5 confinement–deconfinement transition for the $\Gamma$6 toric code with $\Gamma$7-noise, where $\Gamma$8 drops from $\Gamma$9 to $e$0 (Lu, 2024).

Boundary criticality between distinct SPT phases can also be controlled by a single edge parameter. In the $e$1 chain interpolating between $e$2 and $e$3,

$e$4

the bulk is critical at $e$5, while the boundary parameter $e$6 selects two stable $e$7D boundary phases: an odd-sublattice symmetry-broken phase for $e$8 and an even-sublattice symmetry-broken phase for $e$9. At $A_v$0 there is a direct non-Landau boundary transition with $A_v$1 and boundary order-parameter exponent $A_v$2, interpreted as a $A_v$3D deconfined quantum critical point (Prembabu et al., 2022).

A complementary mathematical description is available for boundary transitions in $A_v$4D $A_v$5 topological order. The self-dual critical boundary between the $A_v$6-condensed and $A_v$7-condensed boundaries is described by an enriched fusion category obtained via “topological Wick rotation.” The corresponding boundary Hamiltonian

$A_v$8

has a self-dual critical choice $A_v$9, whose continuum limit is the $m$0-parafermion CFT of Fateev–Zamolodchikov (Lu et al., 2022).

These results establish that boundary transitions in quantum topological matter need not be reducible to ordinary surface ordering. They may instead involve anyon condensation, mixed-state entanglement loss, projective-edge incompatibility, or enriched categorical data.

3. Open boundaries, non-Bloch topology, and boundary-localized modes

Open-system boundary control is especially explicit in the SSH chain with leads. When a finite SSH chain is coupled at both ends to semi-infinite undimerized chains, integrating out the leads gives an energy-dependent effective Hamiltonian

$m$1

with $m$2 and $m$3. As the boundary coupling $m$4 grows, the system passes through three regimes. For very small coupling the original SSH edge states survive; in the intermediate region those states broaden into the continuum and the mid-gap density of states is suppressed; for very large coupling the boundary sites lock to the lead sites and the surviving subchain of length $m$5 undergoes a phase reversal, producing “phase-inverted edge states” localized on sites $m$6 and $m$7 rather than $m$8 and $m$9 (Bissonnette et al., 2023).

Non-Hermitian generalized boundary conditions extend this logic beyond the periodic/open dichotomy. In the Hatano–Nelson setting, boundary parameters $B_p$0 interpolate continuously between PBC and OBC, and the resulting single-particle states are superpositions $B_p$1 with $B_p$2. The generalized Brillouin zone consists of the contours traced by $B_p$3 and $B_p$4, and a non-Bloch winding number

$B_p$5

jumps when the GBZ branches touch. In the simplest class with $B_p$6, $B_p$7, and $B_p$8, the exceptional points occur at $B_p$9, with $g$0 (Verma et al., 2023).

A related, more constrained phenomenon is exceptional-point locking in chiral non-Hermitian lattices. In the extended non-Hermitian SSH chain with

$g$1

point-gap transitions under PBC and real-line-gap transitions under OBC are generally distinct. Along an exceptional-point-constrained manifold, however, the Bloch spectrum remains pinned to a zero-energy degeneracy, and the two transition criteria coincide. In the analytically tractable limit $g$2, the EP-constrained manifold is $g$3 or $g$4, while the OBC generalized Brillouin zone is the circle $g$5 (Wang et al., 26 Mar 2026).

Boundary control also resolves bulk–boundary mismatch in bosonic non-Hermitian dynamics. In a dimerized antiferromagnetic chain, linear spin-wave theory yields a non-Hermitian dynamic matrix $g$6. Conventional Bloch invariants vanish, yet finite chains host sublattice-polarized magnon edge modes. Replacing $g$7 by $g$8 defines a non-Bloch dynamic matrix $g$9 and winding number

$4^g$0

A boundary perturbation $4^g$1 then drives the edge states into and out of the bulk spectrum. For typical parameters $4^g$2, $4^g$3, $4^g$4, the critical values are $4^g$5 and $4^g$6, with $4^g$7 changing between $4^g$8 and $4^g$9 (Debnath et al., 8 Jun 2026).

Boundary modes need not even be tied to Chern topology. In six-band toy models of topological skyrmion phases, the skyrmion number

$b$00

can jump without closing the minimum direct bulk energy gap. The defining condition for the type-II transition is instead $b$01, while $b$02 remains finite. In slab geometry this produces overlapping, exponentially localized edge bands that render the strip gapless even when the total Chern number is zero (Ay et al., 2023).

Taken together, these works show that boundary-sensitive topology in open and non-Hermitian systems is governed by effective self-energies, exceptional points, generalized Brillouin zones, and boundary-localized potentials as much as by conventional Bloch invariants.

4. Interfacial topology in crystalline and polycrystalline materials

In polycrystals, the boundary itself is an atomically structured thermodynamic object. In $b$03 symmetric-tilt grain boundaries of Ti, Fe segregation stabilizes icosahedral cages consisting of a central column of Fe atoms at interstitial positions between successive basal $b$04 planes and a shell of twelve Ti atoms arranged as two staggered five-fold rings rotated by $b$05. Increasing Fe excess produces a hierarchy of grain-boundary phases: the clean “ABC” phase, a single-cage phase, a double-cage phase, triple-cage and higher-order clusters, and a layered-cage phase. Semi-grand canonical MD/MC simulations show hysteresis and metastability at the same chemical potential, confirming first-order transitions. The excess solute is quantified by

$b$06

and discontinuous jumps in $b$07 indicate the transitions (Devulapalli et al., 2024).

A different grain-boundary topological transition arises from the unbinding of disconnections. In the coarse-grained model of a GB as a line hosting line defects with Burgers vector $b$08 and step height $b$09, the RG variables

$b$10

obey

$b$11

The separatrix is at $b$12, giving a Kosterlitz–Thouless transition between a bound-dipole, smooth GB phase and an unbound, rough GB phase. In the sparse limit the transition temperature is

$b$13

The predicted consequences include abrupt changes in GB mobility, GB sliding, roughening, grain-growth stagnation, abnormal grain growth, and superplasticity (Chen et al., 2020).

Topological changes in grain-boundary networks during grain growth require explicit enumeration and selection rules once general boundary energies are allowed. For the five-grain junction prototype, graph-search on strata adjacency yields $b$14 inequivalent circuits for new triple lines and $b$15 valid surface insertions. Candidate insertions are compared by the instantaneous energy-dissipation rate $b$16, and the transition with the largest positive $b$17 is selected. The reported five-grain example shows near-degeneracy between conventional and “exceptional” transitions, with Digon I lying within $b$18–$b$19 of the leading mode under some geometries (Eren et al., 2021).

Wall-induced morphology changes in thin films fit the same interfacial logic. In a binary mixture with antisymmetric wall energy $b$20, slight energetic wall bias nucleates boundary layers that create a horizontal bilayer, while partial wetting with $b$21 can destabilize the bilayer into self-replicating trapezoidal vertical stripes. The thin-film limit yields

$b$22

on a moving interval, with stripe width fixed by area conservation and reported as $b$23 for small $b$24 (Hennessy et al., 2014).

These materials examples broaden the meaning of “topological transition.” Here the topology is not primarily a band invariant; it is the topology of interfacial motifs, defect gases, or boundary networks, controlled by local chemistry, elasticity, and capillarity.

5. Experimental realizations and diagnostics

Smooth boundaries between topologically distinct one-dimensional photonic quasicrystals provide a direct probe of bulk phase transitions. In coupled waveguides with off-diagonal Harper or Fibonacci modulation, the deformation region is defined by

$b$25

with $b$26 varying slowly across a length $b$27. The generalized return probability

$b$28

detects localized subgap states in the deformation zone. For topologically distinct quasicrystals, two peaks per large gap appear; for topologically equivalent Harper and Fibonacci quasicrystals, $b$29 is flat and all gaps remain open throughout the deformation (Verbin et al., 2012).

Mechanical metamaterials realize a closely related boundary-driven band inversion. In the cylindrical granular chain, the contact angle $b$30 controls the linearized stiffness

$b$31

An infinite dimer chain with alternating $b$32 and $b$33 has the standard acoustic and optical bands; when $b$34 crosses $b$35, the band gap closes and reopens with inverted Zak phase. For $b$36, a finite chain supports a boundary mode at

$b$37

and joining two chains of opposite topology produces interface modes that were measured by laser Doppler vibrometry (Chaunsali et al., 2017).

At the atomic scale, direct microscopy and spectroscopy identify boundary-controlled transitions in crystalline interfaces. In Ti films containing $b$38 wt\% Fe, HAADF-STEM together with atomic-scale EDX/EELS confirmed Fe enrichment at the centers of icosahedral grain-boundary cages, while GRIP structure prediction and hybrid MD/MC simulations reproduced the stepwise appearance of single-, double-, and layered-cage phases (Devulapalli et al., 2024).

In correlated fermionic systems, the diagnostics are field-theoretic and numerical rather than spectroscopic. For a two-dimensional time-reversal-invariant topological superconductor with open boundaries, determinant quantum Monte Carlo on cylinders $b$39 and two-loop renormalization group analysis reveal ordinary, special, and extraordinary boundary transitions. At the special point, the boundary boson and boundary Majorana correlators give $b$40 and $b$41 in simulation, while the two-loop RG yields $b$42 and $b$43, identifying a boundary Gross–Neveu–Yukawa fixed point (Ge et al., 6 Oct 2025).

Across platforms, the observable is almost always localized: deformation-zone LDOS, interface-mode velocity profile, boundary order parameter, negativity spectrum, open-loop expectation value, or interfacial excess. This suggests that experimental access to boundary-controlled transitions is often better than access to the corresponding bulk invariant.

6. Conceptual issues, misconceptions, and open questions

A common misconception is that every topological transition must be diagnosed by a bulk direct-gap closing. Several of the cited works explicitly show otherwise. Type-II skyrmion transitions change the skyrmion number while the minimum direct bulk energy gap stays finite and the singular object is instead the vanishing of $b$44 at an isolated momentum point (Ay et al., 2023). In generalized-boundary non-Hermitian systems, the relevant transition can be a touching of generalized-momentum contours at an exceptional point rather than a conventional Bloch-band closure (Verma et al., 2023). In mixed-state toric-code problems, the transition concerns the destruction of topological entanglement negativity under boundary decoherence, not the disappearance of the bulk stabilizer order itself (Lu, 2024).

A second misconception is that a single invariant always suffices. In the toric code with varying boundaries, the topological entanglement entropy stays fixed at $b$45 and therefore fails to distinguish phases that differ only by boundary conditions; the open-loop operator and the ground-state degeneracy do detect the transition (Jamadagni et al., 2020). In percolated SSH chains, the many-body polarization tracks global connectivity while the zero-energy mode count tracks cluster-local topology, producing a “Fractured Topological Region” in which $b$46 but $b$47 (Mondal et al., 2023).

A third issue concerns bulk–boundary correspondence. In non-Hermitian magnonics, conventional Bloch invariants vanish although open chains host boundary-localized modes, and the mismatch is repaired only in a non-Bloch framework (Debnath et al., 8 Jun 2026). In chiral non-Hermitian SSH models, periodic-boundary point-gap transitions and open-boundary line-gap transitions are generally decoupled, but they become locked on exceptional-point-constrained manifolds (Wang et al., 26 Mar 2026). This suggests that the relevant “bulk” object may itself depend on the admissible boundary condition.

Several open questions are stated explicitly in the literature. For Fe-segregated Ti grain boundaries, open problems include the generality of icosahedral GB phases in other alloy systems, the kinetics of cage nucleation and growth under non-equilibrium processing, and the effect of external fields on the GB phase diagram (Devulapalli et al., 2024). For boundary decoherence in topological order, the approach is suggested to extend to stabilizer codes such as fracton codes (Lu, 2024). For boundary deconfined criticality, extensions to other cyclic groups, time-reversal invariants, and $b$48D transitions are proposed (Prembabu et al., 2022). For non-Hermitian and wave-based systems, the generalized-boundary and non-Bloch frameworks indicate that boundary engineering can become a design principle rather than a perturbation (Verma et al., 2023, Bissonnette et al., 2023).

In aggregate, boundary-controlled topological transitions form a heterogeneous but coherent research area. The unifying idea is that the boundary may itself carry the control manifold, the critical degrees of freedom, and the operational invariant.