Tasaki Index in Quantum Many-Body Systems

• Tasaki Index is a family of context-dependent invariants that characterize quantum many-body systems, notably differentiating trivial chains from nontrivial Haldane phases in spin-1 models.
• The index is operationally defined via matrix product state formulations, edge-state fractionalization, and entanglement spectra, providing clear diagnostic criteria for symmetry-protected phases.
• Formulations extend to Tasaki-type criteria in flat-band ferromagnetism and hidden symmetry sectors in Kennedy–Tasaki duality, linking representation data with experimental observables.

The supplied literature suggests that the expression Tasaki index is not a universally standardized scalar, but a context-dependent designation for index-like structures associated with Tasaki’s constructions in quantum many-body physics. In the spin-1 Affleck–Kennedy–Lieb–Tasaki chain, it denotes a $\mathbb{Z}_2$ topological invariant characterizing the nontrivial Haldane symmetry-protected topological phase. In later settings built on Tasaki’s methods, the same idea is represented by flat-band ferromagnetism criteria, symmetry-sector labels exposed by the Kennedy–Tasaki transformation, or boundary-state invariants such as defect labels and Affleck–Ludwig boundary entropy (Lu et al., 2015, Liu et al., 2019, Yang et al., 2023, Zhang et al., 18 Aug 2025).

1. AKLT origin and the Haldane-phase index

In the AKLT setting, the Tasaki index is a topological index for the Haldane phase realized by the spin-1 chain with parent Hamiltonian

$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$

with $J>0$. This Hamiltonian is equivalent to a sum of projectors onto the total spin-2 sector on each bond, and its unique ground state is the AKLT state, a representative of the Haldane phase (Lu et al., 2015).

The AKLT ground state is exactly described by a spin-1 matrix-product state,

$$|\Psi_{\mathrm{AKLT}}\rangle = \sum_{\{s_i\}} \mathrm{Tr}\left( \mathbf{A}^{[s_1]}\mathbf{A}^{[s_2]}\cdots \mathbf{A}^{[s_L]} \right) |s_1 s_2 \cdots s_L\rangle,$$

where $s_i=-1,0,+1$ and $\mathbf{A}^{[s]}$ are $2\times 2$ matrices. The MPS structure has a direct physical interpretation: each physical spin-1 is built from two virtual spin-$\tfrac12$ degrees of freedom symmetrized on the site, neighboring sites share a virtual singlet, and an open chain leaves one unpaired virtual spin-$\tfrac12$ at each end. This bulk fractionalization into edge spin-$\tfrac12$ modes is the central diagnostic of the nontrivial phase.

Within this formulation, the index distinguishes a trivial spin-1 chain from the AKLT/Haldane chain. The trivial case has no protected edge spin-$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$0 and a virtual space transforming linearly under the protecting symmetry. The AKLT/Haldane case has protected edge spin-$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$1 modes and a nontrivial projective action on the virtual space. The associated phase is protected by symmetries such as $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$2 spin rotation, time reversal, or the dihedral $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$3 subgroup.

2. Equivalent formulations in MPS, edges, and entanglement

A standard formulation identifies the Tasaki index with the projective representation carried by the virtual MPS degrees of freedom. For the AKLT state, the virtual space transforms as a spin-$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$4 doublet, so the symmetry is realized projectively rather than linearly. In modern language, this is a $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$5 topological invariant that distinguishes the nontrivial Haldane phase from a trivial phase (Lu et al., 2015).

An equivalent formulation appears in the edge and entanglement structure. Cutting the chain leaves two effective spin-$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$6 edge degrees of freedom. For a segment of length $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$7, the reduced density matrix has eigenvalues

$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$8

In the long-segment limit $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$9, all four eigenvalues approach $J>0$0, producing the characteristic fourfold structure associated with two cut-induced spin-$J>0$1 modes. In this sense, the entanglement spectrum is an operational representation of the same index.

The same phase can also be described through nonlocal string order or a $J>0$2 Berry phase. The detailed discussion explicitly notes these as equivalent formulations, although they are not re-derived there. A concise operational statement is: the index asks whether the ground-state MPS carries half-integer virtual spin and therefore exhibits protected edge spin-$J>0$3 modes and the corresponding entanglement degeneracy. If so, the phase is nontrivial.

3. Entanglement Hamiltonians and infinite-randomness criticality

The AKLT realization of the Tasaki index becomes especially explicit in entanglement Hamiltonians. For a bipartition $J>0$4, the reduced density matrix is written as

$J>0$5

For a single open segment of length $J>0$6, the reduced density matrix has exactly the four eigenvalues above, and the corresponding entanglement Hamiltonian is

$J>0$7

with

$J>0$8

Here $J>0$9 are the spin-$$|\Psi_{\mathrm{AKLT}}\rangle = \sum_{\{s_i\}} \mathrm{Tr}\left( \mathbf{A}^{[s_1]}\mathbf{A}^{[s_2]}\cdots \mathbf{A}^{[s_L]} \right) |s_1 s_2 \cdots s_L\rangle,$$0 edge variables. The index is thus realized in entanglement space as a pair of interacting fractionalized edge modes (Lu et al., 2015).

Under an extensive bipartition into alternating segments, the short correlation length of the AKLT state implies an effective nearest-neighbor spin-$$|\Psi_{\mathrm{AKLT}}\rangle = \sum_{\{s_i\}} \mathrm{Tr}\left( \mathbf{A}^{[s_1]}\mathbf{A}^{[s_2]}\cdots \mathbf{A}^{[s_L]} \right) |s_1 s_2 \cdots s_L\rangle,$$1 entanglement Hamiltonian,

$$|\Psi_{\mathrm{AKLT}}\rangle = \sum_{\{s_i\}} \mathrm{Tr}\left( \mathbf{A}^{[s_1]}\mathbf{A}^{[s_2]}\cdots \mathbf{A}^{[s_L]} \right) |s_1 s_2 \cdots s_L\rangle,$$2

with $$|\Psi_{\mathrm{AKLT}}\rangle = \sum_{\{s_i\}} \mathrm{Tr}\left( \mathbf{A}^{[s_1]}\mathbf{A}^{[s_2]}\cdots \mathbf{A}^{[s_L]} \right) |s_1 s_2 \cdots s_L\rangle,$$3 set by segment lengths. A uniform extensive bipartition with equal even segment lengths yields an effective clean critical spin-$$|\Psi_{\mathrm{AKLT}}\rangle = \sum_{\{s_i\}} \mathrm{Tr}\left( \mathbf{A}^{[s_1]}\mathbf{A}^{[s_2]}\cdots \mathbf{A}^{[s_L]} \right) |s_1 s_2 \cdots s_L\rangle,$$4 chain with central charge $$|\Psi_{\mathrm{AKLT}}\rangle = \sum_{\{s_i\}} \mathrm{Tr}\left( \mathbf{A}^{[s_1]}\mathbf{A}^{[s_2]}\cdots \mathbf{A}^{[s_L]} \right) |s_1 s_2 \cdots s_L\rangle,$$5, corresponding to the critical line where the Haldane phase collapses into a trivial dimer phase. A random extensive bipartition with even-length segments distributed as

$$|\Psi_{\mathrm{AKLT}}\rangle = \sum_{\{s_i\}} \mathrm{Tr}\left( \mathbf{A}^{[s_1]}\mathbf{A}^{[s_2]}\cdots \mathbf{A}^{[s_L]} \right) |s_1 s_2 \cdots s_L\rangle,$$6

induces a coupling distribution

$$|\Psi_{\mathrm{AKLT}}\rangle = \sum_{\{s_i\}} \mathrm{Tr}\left( \mathbf{A}^{[s_1]}\mathbf{A}^{[s_2]}\cdots \mathbf{A}^{[s_L]} \right) |s_1 s_2 \cdots s_L\rangle,$$7

which is exactly the fixed-point distribution of the random-singlet phase of the random spin-$$|\Psi_{\mathrm{AKLT}}\rangle = \sum_{\{s_i\}} \mathrm{Tr}\left( \mathbf{A}^{[s_1]}\mathbf{A}^{[s_2]}\cdots \mathbf{A}^{[s_L]} \right) |s_1 s_2 \cdots s_L\rangle,$$8 Heisenberg chain.

The nested entanglement entropy of the ground state of $$|\Psi_{\mathrm{AKLT}}\rangle = \sum_{\{s_i\}} \mathrm{Tr}\left( \mathbf{A}^{[s_1]}\mathbf{A}^{[s_2]}\cdots \mathbf{A}^{[s_L]} \right) |s_1 s_2 \cdots s_L\rangle,$$9 obeys

$s_i=-1,0,+1$0

with fitted

$s_i=-1,0,+1$1

This realizes the infinite-randomness fixed point directly in entanglement space. The crucial point for the index is structural: the random-singlet network exists because every cut exposes a spin-$s_i=-1,0,+1$2 mode dictated by the nontrivial AKLT/Haldane index.

4. Tasaki-type criteria in flat-band ferromagnetism

In the SU$s_i=-1,0,+1$3 Hubbard model on the Tasaki lattice, the paper does not define a literal scalar called the Tasaki index. Instead, it implements what the detailed discussion calls a Tasaki-type criterion for flat-band ferromagnetism. The Tasaki lattice is a decorated hypercubic lattice obtained by adding one extra site in the middle of each nearest-neighbor bond of an undecorated hypercubic lattice $s_i=-1,0,+1$4. The number of central sites is $s_i=-1,0,+1$5, and this $s_i=-1,0,+1$6 is exactly the degeneracy of the lowest flat band. For hopping amplitudes $s_i=-1,0,+1$7 on the undecorated lattice and $s_i=-1,0,+1$8 between central and decorated sites, the flat-band tuning condition is

$s_i=-1,0,+1$9

where $\mathbf{A}^{[s]}$0 is the coordination number of the undecorated lattice. The localized flat-band states are the trapping-cell states $\mathbf{A}^{[s]}$1, and the construction satisfies Tasaki’s quasi-locality and local connectivity conditions (Liu et al., 2019).

The rigorous structure of the ground-state manifold is formulated through two constraints on the trapping-cell expansion of a ground state. First, if two colors occupy the same trapping cell, the corresponding coefficient vanishes:

$\mathbf{A}^{[s]}$2

Second, if two configurations have the same cluster decomposition and the same number of particles of each color within every cluster, then their coefficients are equal:

$\mathbf{A}^{[s]}$3

These conditions force each connected cluster to carry the fully symmetric irreducible representation of $\mathbf{A}^{[s]}$4, with dimension

$\mathbf{A}^{[s]}$5

For $\mathbf{A}^{[s]}$6, every ground state is a linear combination of such ferromagnetic-cluster states. At $\mathbf{A}^{[s]}$7, meaning $\mathbf{A}^{[s]}$8, the entire trapping-cell graph forms a single cluster, and the ground state is rigorously ferromagnetic.

The same work maps the partially filled problem to Pauli-correlated percolation, with configuration weight

$\mathbf{A}^{[s]}$9

This yields an effective percolation-based diagnostic for ferromagnetism. In one dimension, the percolation transition occurs only at $2\times 2$0. In two dimensions, the transition is first-order like, with a phase-separated regime between $2\times 2$1 and $2\times 2$2, and the transition range increases with $2\times 2$3.

Model	$2\times 2$4	$2\times 2$5
SU(3)	$2\times 2$6	$2\times 2$7
SU(4)	$2\times 2$8	$2\times 2$9
SU(10)	$\tfrac12$0	$\tfrac12$1

For comparison, standard uncorrelated site percolation on the corresponding square-like connectivity has threshold $\tfrac12$2. The detailed discussion therefore proposes, as a plausible distillation rather than a literal paper definition, either a filling-based Tasaki index

$\tfrac12$3

or a percolation-based Tasaki index given by the probability or fraction of sites in the largest ferromagnetic cluster in the PCP model. In this usage, the “index” is a constructive or diagnostic criterion for flat-band ferromagnetism rather than an SPT invariant.

5. Kennedy–Tasaki duality and hidden symmetry sectors

A third usage arises from the Kennedy–Tasaki transformation in spin-1 chains. Here again the phrase Tasaki index is not explicit, but the detailed discussion identifies the relevant index-like quantities as symmetry quantum numbers and hidden conserved charges. The original model is the spin-1 XXZ chain with single-ion anisotropy,

$\tfrac12$4

which has internal $\tfrac12$5 symmetry, a $\tfrac12$6 subgroup generated by $\tfrac12$7 rotations, and bond-centered inversion. For spin 1, the on-site $\tfrac12$8-rotation operators are

$\tfrac12$9

The natural sector labels are $\tfrac12$0, where $\tfrac12$1 is total magnetization and $\tfrac12$2 are the eigenvalues of global $\tfrac12$3, $\tfrac12$4, and inversion (Yang et al., 2023).

The Kennedy–Tasaki unitary is

$\tfrac12$5

with $\tfrac12$6 and $\tfrac12$7. It maps the original short-range Hamiltonian to another short-range Hamiltonian $\tfrac12$8. In the dual model, the visible on-site symmetry is only $\tfrac12$9, while the original $\tfrac12$0 becomes a hidden non-local symmetry generated by

$\tfrac12$1

The dual sectors are therefore labeled visibly by $\tfrac12$2, with an additional hidden $\tfrac12$3 charge that is conserved but nonlocal and impractical for numerical sector decomposition.

This hidden symmetry has direct spectral consequences. In the original model, GOE level-spacing statistics are obtained once the full symmetry, including $\tfrac12$4, is resolved. If one uses only the coarser $\tfrac12$5 labels, different $\tfrac12$6 sectors are mixed and level repulsion is lost. In the dual model, visible $\tfrac12$7 sectors likewise show non-GOE statistics because the hidden $\tfrac12$8 charge remains unresolved. When perturbations such as $\tfrac12$9, single-site defects $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$00 or $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$01, or a random field $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$02 are added so as to break the hidden symmetry, GOE statistics are restored. In this setting, a Tasaki-type index is best understood as the set of visible $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$03 and hidden $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$04 quantum numbers that organize hidden order and determine the correct symmetry-sector decomposition.

6. SO$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$05 generalizations and conformal boundary data

The most recent generalization again does not use the phrase explicitly, but it identifies several quantized quantities that naturally play the role of AKLT/Tasaki indices. The lattice model is the SO$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$06-symmetric bilinear–biquadratic chain

$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$07

It has a Uimin–Lai–Sutherland point at

$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$08

with low-energy limit $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$09 WZW, a second integrable point at

$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$10

and a special MPS point

$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$11

where the ground states are exactly known SO$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$12 AKLT states (Zhang et al., 18 Aug 2025).

These SO$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$13 AKLT states are built from Gamma matrices satisfying $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$14. The crucial structural datum is that the physical degrees of freedom transform in the vector representation of $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$15, while the virtual MPS space carries a spinor representation of $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$16. For odd $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$17 there is a unique ground state; for even $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$18 there are two ground states obtained as symmetric and antisymmetric combinations of two non-injective MPS states. The detailed discussion identifies the virtual spinor representation, the associated edge representation, and the edge-state degeneracy $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$19 as natural SPT-type index data.

At the continuum level, the work uses the conformal embedding $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$20 to construct non-Cardy boundary states. For odd $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$21, the relevant topological defect line is $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$22, yielding a boundary state with

$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$23

For even $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$24, the relevant defects are $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$25, with

$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$26

The overlaps of the SO$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$27 AKLT MPS states with the $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$28 ULS ground state have asymptotic form

$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$29

and the exact calculation gives

$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$30

The detailed discussion therefore presents the defect labels $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$31 or $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$32, the virtual spinor representation, and the Affleck–Ludwig boundary entropy $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$33 as the most natural index-like invariants in this generalization.

Taken together, these formulations show that the Tasaki index is best regarded not as a single immutable definition but as a family of discrete or threshold-type invariants tied to Tasaki’s constructions: a $$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$34 SPT index in the spin-1 AKLT chain, a ferromagnetic criterion based on flat-band occupancy and percolation in Tasaki lattices, a set of visible and hidden symmetry charges in Kennedy–Tasaki duality, and representation- or boundary-entropy data in SO$$H_{\mathrm{AKLT}}=\sum_{i=1}^{L} J \left[ \mathbf{s}_i \cdot \mathbf{s}_{i+1} + \frac{1}{3}(\mathbf{s}_i \cdot \mathbf{s}_{i+1})^2 \right],$$35 AKLT generalizations.