Dynamical Bulk-Boundary Correspondence

• Dynamical bulk-boundary correspondence is the conversion of evolving bulk topological, spectral, or dynamical quantities into structured, observable boundary signatures under time-dependent conditions.
• It encompasses diverse frameworks—including operator-algebraic T-duality, quench dynamics with Loschmidt spectra, and spectral Green’s-function methods—each providing unique diagnostic markers.
• These approaches offer practical tools for predicting edge mode behavior and dynamical phase transitions in systems such as topological insulators, SSH chains, and Floquet models.

Searching arXiv for recent and foundational papers on dynamical bulk-boundary correspondence. arXiv search query: "dynamical bulk boundary correspondence topological insulators Loschmidt matrix" Dynamical bulk-boundary correspondence denotes a set of relations in which bulk topological, spectral, or dynamical structures are reflected in boundary observables in settings that are explicitly time-dependent, frequency-resolved, or formulated through crossed-product dynamics. In the literature, the term is used in several closely related ways. In operator $K$-theory, it refers to the identification of the bulk-boundary homomorphism with geometric restriction after T-duality; in quench dynamics, it refers to the synchronization between bulk dynamical quantum phase transitions and boundary signatures in the Loschmidt spectrum; in spectral and non-Hermitian settings, it refers to frequency- or rate-resolved correspondences between bulk invariants and boundary responses. Across these formulations, the common theme is that a boundary contribution is not an auxiliary finite-size effect, but a structured diagnostic of bulk dynamics (Hannabuss et al., 2016, Sedlmayr et al., 2017, Tamura et al., 2021).

1. Conceptual scope and principal formulations

The equilibrium bulk-boundary correspondence links bulk topological invariants to protected boundary states. The dynamical generalizations surveyed here retain that organizing principle but replace static band data by objects such as crossed-product exact sequences, Loschmidt amplitudes, Floquet return maps, Green’s functions, generalized Brillouin zones, or Liouvillian rapidities. The bulk datum can therefore be a $K$-theoretic boundary map, a Fisher-zero crossing, a unitary-loop winding, a spectral winding-like invariant, or a non-Bloch topological index; the boundary datum can be a restriction map, a boundary return rate, near-zero Loschmidt eigenvalues, anomalous Floquet edge or hinge modes, odd-frequency boundary response, or dynamically dominant non-Hermitian boundary modes (Hannabuss et al., 2016, Sedlmayr et al., 2017, Vu, 2021).

A first important distinction is between geometric/operator-algebraic and quench-dynamical usages. In the former, one starts from a bulk algebra and a boundary algebra related by a Toeplitz or Pimsner–Voiculescu extension, and T-duality turns the abstract bulk-boundary map into geometric restriction (Hannabuss et al., 2016). In the latter, one starts from a many-body state prepared as the ground state of an initial Hamiltonian and evolved by a post-quench Hamiltonian; nonanalyticities of the return rate coincide with boundary-localized structures in the Loschmidt matrix spectrum (Sedlmayr et al., 2017). A third usage appears in spectral Green’s-function language, where the correspondence is promoted to the frequency domain through relations such as

$z F(z)=w(z),$

and remains valid with disorder and dynamical self-energies, even without chiral symmetry (Tamura et al., 2021).

These formulations are not identical. A plausible implication is that “dynamical bulk-boundary correspondence” should be understood as a family resemblance term rather than a single universal theorem. What unifies the family is the conversion of bulk information into a boundary-resolved quantity by an explicitly dynamical construction.

2. Operator-algebraic bulk-boundary maps and T-duality

In the noncommutative formulation, the boundary algebra is a separable real or complex $C^*$-algebra $\mathcal B$ equipped with a translation action parallel to the boundary and an automorphism in the normal direction. In the simplest codimension-one setting, the bulk algebra is the crossed product

$$A_{\mathrm{bulk}}=\mathcal B\rtimes_\alpha \mathbb Z,$$

and the half-space geometry is encoded by the Toeplitz extension

$$0\to \mathcal B\otimes \mathcal K \xrightarrow{\iota} \mathcal T_\alpha \xrightarrow{\pi} \mathcal B\rtimes_\alpha \mathbb Z \to 0.$$

Its associated Pimsner–Voiculescu exact sequence contains the connecting map

$$\partial:K_i(\mathcal B\rtimes_\alpha \mathbb Z)\to K_{i-1}(\mathcal B),$$

which is the bulk-boundary homomorphism in momentum-space language (Hannabuss et al., 2016).

The geometric counterpart is a mapping-torus construction. With $U\simeq \mathbb R$, $V\simeq \mathbb R^{d-1}$, and $K$0, one defines

$K$1

with geometric short exact sequence

$K$2

Here $K$3 is evaluation at $K$4, i.e. geometric restriction to the boundary. After crossing with the amenable parallel-translation group $K$5, one obtains the physical extension whose $K$6-theory long exact sequence is the PV sequence for the T-dual bulk algebra (Hannabuss et al., 2016).

The central result is that T-duality trivializes the abstract boundary map. In the paper’s formulation, Paschke’s map equals Connes–Thom composed with Green’s imprimitivity, and the PV connecting map $K$7 is intertwined with the geometric restriction map $K$8. Equivalently, the diagram

$K$9

commutes, so T-duality converts the dynamical PV bulk-boundary map into geometric restriction (Hannabuss et al., 2016).

The generality is unusually broad. The result holds in arbitrary spatial dimension, in both the real and complex cases, and in the presence of disorder, magnetic fields, and $z F(z)=w(z),$0-flux. In the parametrized $z F(z)=w(z),$1-flux setting, the bulk-boundary homomorphism is again trivialized by T-duality, now in twisted $z F(z)=w(z),$2-theory. This provides a unified geometric picture relevant both to string-theoretic T-duality and to condensed-matter systems such as the quantum Hall effect and topological insulators with defects (Hannabuss et al., 2016).

3. Quenches, Loschmidt matrices, and boundary return rates

In quench problems, one prepares the initial state $z F(z)=w(z),$3 as the ground state of $z F(z)=w(z),$4 and evolves it with $z F(z)=w(z),$5. The Loschmidt amplitude and return rate are

$z F(z)=w(z),$6

or, in equivalent conventions, $z F(z)=w(z),$7. Dynamical quantum phase transitions occur when $z F(z)=w(z),$8 or its derivatives become nonanalytic at real critical times $z F(z)=w(z),$9, generated by Fisher zeros crossing the real-time axis (Sedlmayr et al., 2017, Masłowski et al., 2023).

For free fermions, the decisive object is the Loschmidt matrix

$C^*$0

with $C^*$1. The Loschmidt amplitude factorizes as

$C^*$2

where $C^*$3 are eigenvalues of $C^*$4. The bulk-boundary question is then reformulated spectrally: whether bulk nonanalyticities in the return rate are accompanied by boundary-localized zero or near-zero modes of the nonunitary matrix $C^*$5 (Sedlmayr et al., 2017, Masłowski et al., 15 Aug 2025).

The one-dimensional prototype is the quenched SSH chain. For open chains,

$C^*$6

and the critical times at which the bulk rate $C^*$7 develops cusps coincide with sudden changes in the boundary contribution $C^*$8. For trivial-to-topological quenches, two eigenvalues of the dynamical Loschmidt matrix approach zero exponentially with system size and remain near zero over extended intervals, so that

$C^*$9

tracks the finite-size boundary piece. The same times are tied to the periodic creation of long-range entanglement between the edges (Sedlmayr et al., 2017).

The mechanism generalizes to other one-dimensional symmetry-protected phases. In a long-range Kitaev chain, quenches between phases with different winding numbers produce multiple critical momenta and multiple critical times, while the number of near-zero Loschmidt eigenvalues matches the increase in the number of edge-state pairs. This establishes the correspondence at the level of edge-mode counting in the Loschmidt spectrum rather than solely at the level of cusps in the bulk return rate (Sedlmayr et al., 2017).

4. Two-dimensional and higher-order dynamical correspondences

In two dimensions, the bulk singularity structure is typically more elaborate. Fisher zeros generically form areas in the complex-time plane, and the sharp thermodynamic diagnostic is often the derivative of the return rate,

$\mathcal B$0

Nevertheless, open geometries again reveal a boundary-sensitive sector of the Loschmidt matrix spectrum (Masłowski et al., 2023, Masłowski et al., 15 Aug 2025).

For higher-order topological insulators, the boundary signature is corner-localized rather than edge-localized. In the Benalcazar–Bernevig–Hughes model, quenches across the equilibrium phase boundary generate DQPTs and a boundary return-rate contribution extracted by finite-size scaling,

$\mathcal B$1

For quenches into the higher-order topological phase, four eigenvalues of the Loschmidt overlap matrix are pinned to zero between alternating critical times, in correspondence with the four corner modes of the static phase. The paper emphasizes that, contrary to the usual two-dimensional case, the DQPTs appear as cusps in the return rate itself rather than in its derivative, because the Fisher-zero plane collapses to a line (Masłowski et al., 2024).

A generalized four-band HOTI model extends this picture beyond the intrinsic BBH point. There, DQPTs occur for quenches that cross both bulk and boundary gap closings, and the dynamical bulk-boundary correspondence takes a form different from one dimension. In two-corner phases, approximately one pinned near-zero eigenvalue appears in certain intervals between critical regions and then disappears for later intervals; in four-corner phases, approximately three pinned near-zero eigenvalues occur between early critical intervals. No pinned near-zero modes occur outside the critical intervals for quenches into trivial phases (Masłowski et al., 2023).

For first-order two-dimensional topological matter, the boundary signature need not consist of pinned zero modes. In a class D model based on the 2D Kitaev lattice, the Loschmidt spectrum of open systems develops in-gap bands between successive critical times if the time-evolving Hamiltonian is topologically non-trivial. The return rate admits finite-size decompositions

$\mathcal B$2

for a ribbon, and

$\mathcal B$3

for a square flake. The in-gap bands directly produce the boundary dynamical free energy, with estimates

$\mathcal B$4

depending on geometry and on exponentially small edge-mode contributions (Masłowski et al., 15 Aug 2025).

The resulting phenomenology can be summarized compactly:

Setting	Bulk dynamical object	Boundary signature
1D SPT quench	Fisher-zero crossings, cusps in $\mathcal B$5	Near-zero Loschmidt eigenvalues and jump-like $\mathcal B$6
2D HOTI quench	Critical intervals $\mathcal B$7, discontinuities in $\mathcal B$8 or cusps in $\mathcal B$9	Corner-localized near-zero modes
2D class D quench	Critical regions between successive DQPTs	Dispersing in-gap bands of the Loschmidt matrix
2D many-band BdG quench	Loschmidt spectral-gap closings and critical regions	Edge-orientation-dependent in-gap Loschmidt bands

A misconception corrected by the recent many-band literature is that the correspondence should follow directly from the static Chern numbers of the initial and final phases. In a 2D honeycomb BdG model beyond two-band structure, there is no straightforward correspondence between the equilibrium phases quenched between and the dynamical bulk-boundary correspondence; moreover, the effect can depend on whether the edges are zigzag or armchair, suggesting a possible weak topological variant (Masłowski et al., 9 Jun 2026).

5. Floquet, spectral, and non-Hermitian extensions

Periodic driving provides another major arena. In a periodically driven SSH chain, an adiabatically ramped high-frequency drive creates a chiral mass term in the effective Floquet Hamiltonian, temporarily breaking particle-hole and chiral symmetry, then restores the symmetry after the winding number has changed. The resulting stroboscopic Floquet Hamiltonian carries a nontrivial winding, and open chains exhibit stroboscopic localized edge states exactly when the Floquet winding becomes nontrivial. The same mechanism realizes higher winding phases in extended SSH chains, with two pairs of zero-energy localized edge modes for a stroboscopic $$A_{\mathrm{bulk}}=\mathcal B\rtimes_\alpha \mathbb Z,$$0 phase (Bandyopadhyay et al., 2019).

Anomalous Floquet topology shifts the emphasis from Floquet bands to micromotion. When the $$A_{\mathrm{bulk}}=\mathcal B\rtimes_\alpha \mathbb Z,$$1-gap is open, the return map

$$A_{\mathrm{bulk}}=\mathcal B\rtimes_\alpha \mathbb Z,$$2

defines a unitary loop. Its spacetime winding, rather than the topology of $$A_{\mathrm{bulk}}=\mathcal B\rtimes_\alpha \mathbb Z,$$3, determines anomalous boundary modes at quasienergies $$A_{\mathrm{bulk}}=\mathcal B\rtimes_\alpha \mathbb Z,$$4 and $$A_{\mathrm{bulk}}=\mathcal B\rtimes_\alpha \mathbb Z,$$5. A systematic dynamic bulk-boundary correspondence has been derived for classes BDI, D, DIII, and AII, including inversion-symmetric second-order anomalous Floquet topology, using a dimensional hierarchy and the Atiyah–Hirzebruch spectral sequence to identify the subspaces that carry the relevant topological information (Vu, 2021).

A distinct but related extension is the spectral bulk-boundary correspondence. For a semi-infinite system with Green’s function $$A_{\mathrm{bulk}}=\mathcal B\rtimes_\alpha \mathbb Z,$$6 and bulk Green’s function $$A_{\mathrm{bulk}}=\mathcal B\rtimes_\alpha \mathbb Z,$$7, the canonical relation

$$A_{\mathrm{bulk}}=\mathcal B\rtimes_\alpha \mathbb Z,$$8

equates an edge spectral observable to a bulk winding-like spectral invariant. The formulation survives impurity scattering and dynamical self-energies, regardless of whether the energy spectrum is gapped, and even extends to systems without chiral symmetry. In superconducting nanowires, Kitaev chains, and SSH models, it connects boundary odd-frequency response or sublattice-resolved density of states to bulk spectral flow (Tamura et al., 2021).

In non-Hermitian dynamics, conventional Bloch invariants are insufficient because of the skin effect. In a non-unitary quantum walk, the correct topological data are non-Bloch invariants defined on generalized Brillouin zones, and the domain-wall edge-state count is

$$A_{\mathrm{bulk}}=\mathcal B\rtimes_\alpha \mathbb Z,$$9

The same experiment directly reconstructed topological edge states dynamically by the filtered state

$$0\to \mathcal B\otimes \mathcal K \xrightarrow{\iota} \mathcal T_\alpha \xrightarrow{\pi} \mathcal B\rtimes_\alpha \mathbb Z \to 0.$$0

which isolates the $$0\to \mathcal B\otimes \mathcal K \xrightarrow{\iota} \mathcal T_\alpha \xrightarrow{\pi} \mathcal B\rtimes_\alpha \mathbb Z \to 0.$$1-edge-sector when the quasienergy spectrum is real (Xiao et al., 2019).

For quadratic bosonic Lindbladians, the relevant dynamical criterion is a rate separation rather than a Loschmidt singularity. Writing the damping matrix spectrum in terms of bulk and boundary rapidities, the Liouvillian separation gap

$$0\to \mathcal B\otimes \mathcal K \xrightarrow{\iota} \mathcal T_\alpha \xrightarrow{\pi} \mathcal B\rtimes_\alpha \mathbb Z \to 0.$$2

determines when non-Hermitian boundary modes can be dynamically discerned from bulk skin modes. A positive $$0\to \mathcal B\otimes \mathcal K \xrightarrow{\iota} \mathcal T_\alpha \xrightarrow{\pi} \mathcal B\rtimes_\alpha \mathbb Z \to 0.$$3 isolates a boundary mode as the slowest-decaying or fastest-growing mode, producing telltale signatures in both stable and unstable regimes (Yang et al., 19 Jun 2025).

6. Generality, caveats, and open directions

The strongest rigorous generality currently belongs to the operator-algebraic formulation. There, the correspondence covers arbitrary spatial dimension, real and complex $$0\to \mathcal B\otimes \mathcal K \xrightarrow{\iota} \mathcal T_\alpha \xrightarrow{\pi} \mathcal B\rtimes_\alpha \mathbb Z \to 0.$$4-theory, disorder modeled by $$0\to \mathcal B\otimes \mathcal K \xrightarrow{\iota} \mathcal T_\alpha \xrightarrow{\pi} \mathcal B\rtimes_\alpha \mathbb Z \to 0.$$5 with amenable group action, magnetic $$0\to \mathcal B\otimes \mathcal K \xrightarrow{\iota} \mathcal T_\alpha \xrightarrow{\pi} \mathcal B\rtimes_\alpha \mathbb Z \to 0.$$6-cocycles handled by stabilization, and continuous-trace algebras with $$0\to \mathcal B\otimes \mathcal K \xrightarrow{\iota} \mathcal T_\alpha \xrightarrow{\pi} \mathcal B\rtimes_\alpha \mathbb Z \to 0.$$7-flux. The conceptual message is that the bulk-boundary map is not eliminated by these complications; under T-duality it becomes geometrically transparent (Hannabuss et al., 2016).

The quench literature is broader phenomenologically but less unified structurally. In one dimension, the evidence is sharp: DQPT critical times, near-zero Loschmidt eigenvalues, and boundary return-rate changes coincide. In 2D HOTIs and class D models, the same pattern persists in modified form, but the detailed boundary signature may be pinned zero modes, corner-localized near-zero modes, or dispersing in-gap bands, depending on symmetry class, codimension, and geometry (Sedlmayr et al., 2017, Masłowski et al., 2023, Masłowski et al., 15 Aug 2025).

Several limitations are explicit in the current record. For 2D HOTIs, a general proof of DBBC and its relation to nested Wilson-loop dynamics remains open (Masłowski et al., 2023). In many-band 2D superconductors, the cause and generality of the correspondence remain unclear; a simple rule based on $$0\to \mathcal B\otimes \mathcal K \xrightarrow{\iota} \mathcal T_\alpha \xrightarrow{\pi} \mathcal B\rtimes_\alpha \mathbb Z \to 0.$$8 across the quench fails, and edge orientation can matter qualitatively (Masłowski et al., 9 Jun 2026). This suggests that a full theory may require a specifically non-Hermitian topological treatment of Loschmidt matrices in two dimensions.

Another important caveat is that bulk-edge correspondence itself can fail when the boundary realization changes. In a rotating shallow-water model with odd-viscous regularization, local self-adjoint boundary conditions form a manifold parameterized by von Neumann unitaries $$0\to \mathcal B\otimes \mathcal K \xrightarrow{\iota} \mathcal T_\alpha \xrightarrow{\pi} \mathcal B\rtimes_\alpha \mathbb Z \to 0.$$9, and both correspondence and its violation are typical. The identified mechanism for violations is the conversion of parabolic high-$$\partial:K_i(\mathcal B\rtimes_\alpha \mathbb Z)\to K_{i-1}(\mathcal B),$$0 branches into asymptotically flat ones, which changes the edge index at infinity (Graf et al., 2024). A plausible implication is that dynamical bulk-boundary correspondence in continuum media is inseparable from a precise specification of the boundary operator domain.

Taken together, the modern literature supports a broad but nonuniform conclusion. Dynamical bulk-boundary correspondence is firmly established in several exact and numerically controlled frameworks, yet it is not exhausted by a single invariant recipe. Depending on context, the correspondence is realized as geometric restriction after T-duality, as near-zero or in-gap structure in a Loschmidt spectrum, as a spectral Green’s-function identity, as a Floquet unitary-loop index, or as a rate-separated non-Hermitian boundary mode. The open problem is not whether such correspondences exist, but how far a common formalism can subsume their diverse mechanisms (Hannabuss et al., 2016, Masłowski et al., 9 Jun 2026).