BISS Framework: Topological Phase Dynamics

• The BISS Framework is a methodology that reduces d-dimensional bulk invariants to lower-dimensional winding or Chern numbers defined on band-inversion surfaces.
• It employs quantum quench protocols to measure pseudospin dynamics, enabling direct experimental detection and classification of both first-order and higher-order topological phases.
• Its nested BIS construction and Clifford-algebra decomposition provide a scalable, robust bulk–boundary correspondence essential for characterizing higher-order topological insulators.

The BISS (Band-Inversion Surface and Spin) Framework is a robust and widely implemented dynamical methodology for the classification and detection of topological phases of matter, particularly for systems supporting higher-order topology. Its core principle is the reduction of $d$-dimensional bulk topological invariants to winding or Chern numbers defined on lower-dimensional momentum-space submanifolds known as band-inversion surfaces (BISs). Importantly, the BISS framework generalizes naturally to higher-order topological insulators (HOTIs) via nested BIS constructions, and provides experimentally accessible dynamical protocols—typically quantum quenches—for direct measurement of topological invariants, even in systems lacking explicit symmetry protection.

1. Mathematical Structure and Clifford-Algebra Decomposition

At the foundation of the BISS framework is the representation of the system's Bloch Hamiltonian $H(k)$ as a sum of anticommuting operators from a Clifford algebra: $$H(k) = \sum_{j=1}^J h_j(k)\,\gamma_j,\qquad \{\gamma_j, \gamma_{j'}\} = 2\delta_{jj'}$$ where $k$ is a $d$-dimensional quasimomentum, and $J = d + n$ for $\mathbb{Z}$-type $n$th-order topology. Each $h_j(k)$ corresponds to a momentum-dependent control parameter or mass/spin-orbit term. The minimal $J$ ensures the possibility of defining nested band-inversion surfaces of arbitrary order (Li et al., 2020).

For first-order topology in $H(k)$0 dimensions ($H(k)$1), $H(k)$2 and the topology is captured by mappings $H(k)$3. Higher-order topologies require successive additions of mass terms: for $H(k)$4th-order in $H(k)$5D, a total of $H(k)$6 anticommuting $H(k)$7-matrices are needed (Lei et al., 2022).

2. Hierarchy of Band-Inversion Surfaces and Nested Structure

The defining feature of the BISS framework is the introduction of $H(k)$8th-order BISs, given by the simultaneous vanishing of $H(k)$9 coefficients: $$H(k) = \sum_{j=1}^J h_j(k)\,\gamma_j,\qquad \{\gamma_j, \gamma_{j'}\} = 2\delta_{jj'}$$0 A 1-BIS ($$H(k) = \sum_{j=1}^J h_j(k)\,\gamma_j,\qquad \{\gamma_j, \gamma_{j'}\} = 2\delta_{jj'}$$1) is a $$H(k) = \sum_{j=1}^J h_j(k)\,\gamma_j,\qquad \{\gamma_j, \gamma_{j'}\} = 2\delta_{jj'}$$2-dimensional surface, a 2-BIS is $$H(k) = \sum_{j=1}^J h_j(k)\,\gamma_j,\qquad \{\gamma_j, \gamma_{j'}\} = 2\delta_{jj'}$$3-dimensional, and so on. For characterizing an $$H(k) = \sum_{j=1}^J h_j(k)\,\gamma_j,\qquad \{\gamma_j, \gamma_{j'}\} = 2\delta_{jj'}$$4th-order HOTI, one recursively defines a series of nested BISs:

• The first 2-BIS $$H(k) = \sum_{j=1}^J h_j(k)\,\gamma_j,\qquad \{\gamma_j, \gamma_{j'}\} = 2\delta_{jj'}$$5 identifies boundaries with nontrivial first-order topology (the effective surface BZ).
• One then projects to regions where the first invariant $$H(k) = \sum_{j=1}^J h_j(k)\,\gamma_j,\qquad \{\gamma_j, \gamma_{j'}\} = 2\delta_{jj'}$$6 is nonzero and defines subsequent BISs by zeroing the next pair(s) of coefficients, yielding reduced dimension $$H(k) = \sum_{j=1}^J h_j(k)\,\gamma_j,\qquad \{\gamma_j, \gamma_{j'}\} = 2\delta_{jj'}$$7 at the final step (Li et al., 2020, Lei et al., 2022, Jia et al., 2020).

The process continues until a 1-BIS of dimension $$H(k) = \sum_{j=1}^J h_j(k)\,\gamma_j,\qquad \{\gamma_j, \gamma_{j'}\} = 2\delta_{jj'}$$8 is reached, whose winding or Chern number completes the topological classification. The complete set $$H(k) = \sum_{j=1}^J h_j(k)\,\gamma_j,\qquad \{\gamma_j, \gamma_{j'}\} = 2\delta_{jj'}$$9 uniquely characterizes the $k$0th-order topology.

3. Quantum Quench Protocol and Dynamical Topological Invariants

The central operational component of the BISS framework is a quantum quench protocol encompassing these steps:

1. Initial State Preparation: Fully polarize along some $k$1 direction by applying a large mass term $k$2.
2. Quench: Suddenly substitute with the target Hamiltonian $k$3 at $k$4.
3. Measurement: Compute the long-time average of a chosen pseudospin observable:

$k$5

1. Locating BISs: The zeros of $k$6 precisely trace the BIS defined by $k$7.
2. Topological Invariants: By analyzing how $k$8 changes along the BIS (for various initializations $k$9), one reads out the sign of $d$0 or the winding of $d$1, constructing topological invariants as integrals or discrete sums over BISs (Li et al., 2020, Jia et al., 2020):
   • For a 2-BIS, the surface winding:

   $d$2

• For higher-order hinges or corners, nest further until obtaining the final 1-BIS, for which:

  $d$3

This dynamical scheme is dimension-agnostic and applies both to Hermitian and non-Hermitian systems (via auxiliary Hermitian mapping for the latter) (Lin et al., 29 May 2025).

4. Nested Bulk–Boundary Correspondence and Higher-Order Topology

The BISS framework yields a powerful and generalized bulk–boundary correspondence. In HOTIs, the conventional correspondence between bulk invariants and $d$4-dimensional boundary modes is replaced by a hierarchy:

• The $d$5th-order BIS (a codimension-$d$6 submanifold) marks momentum points where $d$7 mass terms simultaneously vanish, dual to the real-space domain walls (mass domain walls, MDWs) of codimension $d$8 that protect boundary zero modes (Jia et al., 2022).
• Projective reduction and iterative construction ensure that each lower-dimensional subsystem yields an effective first-order Hamiltonian, whose nested winding or Chern invariant governs the emergence of boundary-of-boundary states (e.g., hinge or corner modes). The full nested invariant tuple thus predicts boundary states of the correct codimension, even in the absence of explicit crystal or mirror symmetries (Lei et al., 2022, Jia et al., 2020).

Transitions between different higher-order phases are unified within this momentum-space framework:

• A phase transition of type I (bulk gap closing) occurs if all topological charges cross the BIS at once.
• A type II (boundary gap closing) transition occurs if only a subset of the charges or BISs is involved, corresponding to a boundary-obstructed TPT (Jia et al., 2022).

5. Experimental Realizability and Implementation

The BISS approach is tailored from the outset for experimental accessibility:

• Ultracold atom systems: Spin-orbit coupling terms and mass terms engineered via Raman fields and optical lattices; momentum-resolved measurement and time-of-flight imaging used to extract pseudospin textures and BIS locations.
• Solid-state spin qubits: Momentum mapped onto qubit control parameters, with microwave driving and quench sequences yielding the required dynamics; pseudospin averages measured via standard quantum control and readout (Li et al., 2020).
• Superconducting circuits or photonic networks: Circuit parameters or photon populations correspond to Hamiltonian coefficients; network topology and switching engineer the required terms and quenches.
• Only low-dimensional BIS manifolds are interrogated at each stage, ensuring polynomial (rather than exponential) scaling in measurement effort even for high $d$9 and $J = d + n$0 (Li et al., 2020).

6. Relation to Other Frameworks and Impact

The BISS framework subsumes or is equivalent to several related scheme:

• Conventional BIS methodologies (Floquet topology, interacting/topological symmetry breaking, non-Hermitian systems) are straightforwardly encompassed via higher-order/nested BIS structure or by mapping to auxiliary Hermitian problems (Jia et al., 2020, Lin et al., 29 May 2025).
• Bulk–surface duality and dynamical invariants in lower, higher, or odd dimensions, as developed in multiple works, are special cases or direct consequences of the general nested-BIS dynamical construction (Lei et al., 2022, Ji et al., 2020, Zhang et al., 2018).

Experimentally, the BISS framework has enabled the direct dynamical detection of nontrivial topology in both static and Floquet-engineered systems, allowed for unambiguous measurement of higher-order invariants and their associated transitions, and greatly simplified the protocols required for topological classification by replacing full wavefunction tomography with lower-dimensional spin-averaged measurements. Its dimension-reduction aspect and dynamical robustness have rendered it the standard for HOTI dynamical detection and characterization in contemporary condensed matter experiments (Li et al., 2020).

7. Algorithmic Summary and Practical Computation

The practical computation of higher-order topological invariants within the BISS framework proceeds algorithmically:

1. Write the system Hamiltonian as $J = d + n$1.
2. Define the first 2-BIS $J = d + n$2 by zeros of $J = d + n$3; perform a quench initialized along $J = d + n$4, measure corresponding pseudospin averages, and calculate $J = d + n$5.
3. Restrict to the region where $J = d + n$6, reduce to the effective $J = d + n$7D subsystem, and define the next 2-BIS in the reduced parameter space; repeat the protocol, obtaining $J = d + n$8.
4. Continue recursively until a 1-BIS is reached, at which the sign pattern of the remaining coefficients yields the final Chern/winding invariant $J = d + n$9.
5. The ordered set $\mathbb{Z}$0 provides the topological classification of the $\mathbb{Z}$1th-order phase (Li et al., 2020).

This structure guarantees that every order of topology is interrogated and experimentally accessible using a finite sequence of quantum quench measurements, each involving only lower-dimensional BIS submanifolds and thus scalable to high-dimensional and high-order situations.

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References:

"Direct dynamical characterization of higher-order topological insulators with nested band inversion surfaces" (Li et al., 2020) "Topological classification of Higher-order topological phases with nested band inversion surfaces" (Lei et al., 2022) "Unified characterization for higher-order topological phase transitions" (Jia et al., 2022) "Dynamically characterizing topological phases by high-order topological charges" (Jia et al., 2020) "Quantum simulation for three-dimensional chiral topological insulator" (Ji et al., 2020)