---
title: Tasaki Index in Quantum Many-Body Systems
url: https://www.emergentmind.com/topics/tasaki-index
type: topic
---

# Tasaki Index in Quantum Many-Body Systems

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 [1502.02095] [1901.07004] [2311.14715] [2508.13114].

## 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 [1502.02095].

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-\(\tfrac12\) and a virtual space transforming linearly under the protecting symmetry. The AKLT/Haldane case has protected edge spin-\(\tfrac12\) modes and a nontrivial projective action on the virtual space. The associated phase is protected by symmetries such as \(\mathrm{SO}(3)\) spin rotation, time reversal, or the dihedral \(\mathbb{Z}_2\times \mathbb{Z}_2\) 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-\(\tfrac12\) doublet, so the symmetry is realized projectively rather than linearly. In modern language, this is a \(\mathbb{Z}_2\) topological invariant that distinguishes the nontrivial Haldane phase from a trivial phase [1502.02095].

An equivalent formulation appears in the edge and entanglement structure. Cutting the chain leaves two effective spin-\(\tfrac12\) edge degrees of freedom. For a segment of length \(l\), the reduced density matrix has eigenvalues
$$
\Lambda_0 = \frac{1}{4}\left[1+3\left(-\frac{1}{3}\right)^l\right], \qquad
\Lambda_\alpha = \frac{1}{4}\left[1-\left(-\frac{1}{3}\right)^l\right], \quad \alpha=1,2,3.
$$
In the long-segment limit \(l\to\infty\), all four eigenvalues approach \(\tfrac14\), producing the characteristic fourfold structure associated with two cut-induced spin-\(\tfrac12\) 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 \(\pi\) 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-\(\tfrac12\) 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 \(A\cup B\), the reduced density matrix is written as
$$
\rho_A=\mathrm{Tr}_B\left(|\Psi_{\mathrm{AKLT}}\rangle\langle\Psi_{\mathrm{AKLT}}|\right)\equiv e^{-H_E}.
$$
For a single open segment of length \(l\ge 2\), the reduced density matrix has exactly the four eigenvalues above, and the corresponding entanglement Hamiltonian is
$$
H_E = J(l)\,\boldsymbol{\tau}_L\cdot \boldsymbol{\tau}_R,
$$
with
$$
J(l)=\ln\left[\frac{1+3(-\tfrac13)^l}{1-(-\tfrac13)^l}\right]
\simeq (-1)^l J_0 e^{-l/\xi}, \qquad J_0=4,\quad \xi=\frac{1}{\ln 3}.
$$
Here \(\boldsymbol{\tau}_{L,R}\) are the spin-\(\tfrac12\) edge variables. The index is thus realized in entanglement space as a pair of interacting fractionalized edge modes [1502.02095].

Under an extensive bipartition into alternating segments, the short correlation length of the AKLT state implies an effective nearest-neighbor spin-\(\tfrac12\) entanglement Hamiltonian,
$$
H_E \simeq \sum_i J_i\,\boldsymbol{\tau}_i\cdot \boldsymbol{\tau}_{i+1},
$$
with \(J_i\equiv J(l_i)\) set by segment lengths. A uniform extensive bipartition with equal even segment lengths yields an effective clean critical spin-\(\tfrac12\) chain with central charge \(c=1\), 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
$$
P(l)\sim \frac{1}{\bar l} e^{-(l-l_0)/\bar l},\qquad l\ge l_0,
$$
induces a coupling distribution
$$
P(J)=\frac{1}{\Omega\Gamma}\left(\frac{\Omega}{J}\right)^{1-1/\Gamma},\qquad
\Gamma=\frac{\bar l}{\xi},\quad \Omega=J_0 e^{-l_0/\xi},
$$
which is exactly the fixed-point distribution of the random-singlet phase of the random spin-\(\tfrac12\) Heisenberg chain.

The nested entanglement entropy of the ground state of \(\rho_A\) obeys
$$
\overline{s(l,L_A)}=\frac{\tilde c}{3}\,
\overline{\ln\left[\frac{L_A}{\pi}\sin\left(\frac{\pi l}{L_A}\right)\right]}+\text{const},
$$
with fitted
$$
\tilde c = 0.72\pm 0.02 \approx \ln 2.
$$
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-\(\tfrac12\) mode dictated by the nontrivial AKLT/Haldane index.

## 4. Tasaki-type criteria in flat-band ferromagnetism

In the SU\((N)\) 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 \(V\). The number of central sites is \(|V|=\mathcal{N}\), and this \(\mathcal{N}\) is exactly the degeneracy of the lowest flat band. For hopping amplitudes \(t_1\) on the undecorated lattice and \(t_2\) between central and decorated sites, the flat-band tuning condition is
$$
t_2=\sqrt{c}\,t_1>0,
$$
where \(c\) is the coordination number of the undecorated lattice. The localized flat-band states are the trapping-cell states \(a_{u,\sigma}^\dagger\), and the construction satisfies Tasaki’s quasi-locality and local connectivity conditions [1901.07004].

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:
$$
f(A_1,\cdots,A_N)=0
\quad\text{if}\quad
A_\sigma\cap A_{\sigma'}\neq \emptyset
\ \text{for any}\ \sigma\neq \sigma'.
$$
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:
$$
f(A_1,\cdots,A_N)=f(A_1',\cdots,A_N').
$$
These conditions force each connected cluster to carry the fully symmetric irreducible representation of \(\mathrm{SU}(N)\), with dimension
$$
d_{\mathrm{SU}(N)}(|C|)=\frac{(N+|C|-1)!}{|C|!\,(N-1)!}.
$$
For \(p=n/\mathcal{N}\le 1\), every ground state is a linear combination of such ferromagnetic-cluster states. At \(p=1\), meaning \(n=\mathcal{N}\), 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
$$
W(q)=\prod_{i=1}^{M_q} e^{\mu |C_i|}\, d_{\mathrm{SU}(N)}(|C_i|).
$$
This yields an effective percolation-based diagnostic for ferromagnetism. In one dimension, the percolation transition occurs only at \(p_c=1\). In two dimensions, the transition is first-order like, with a phase-separated regime between \(p_-\) and \(p_+\), and the transition range increases with \(N\).

| Model | \(p_-\) | \(p_+\) |
|---|---:|---:|
| SU(3) | \(0.64(1)\) | \(0.83(0)\) |
| SU(4) | \(0.65(1)\) | \(0.87(7)\) |
| SU(10) | \(0.69(4)\) | \(0.94(5)\) |

For comparison, standard uncorrelated site percolation on the corresponding square-like connectivity has threshold \(p_c\approx 0.5927\). The detailed discussion therefore proposes, as a plausible distillation rather than a literal paper definition, either a filling-based Tasaki index
$$
I_{\mathrm{Tasaki}}=p=\frac{n}{\mathcal{N}},
$$
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,
$$
H=\sum_{i=1}^{L-1}\bigl(S^{x}_{i}S^{x}_{i+1}+S^{y}_{i}S^{y}_{i+1}+\Delta S^{z}_{i}S^{z}_{i+1}\bigr)+D\sum_{i=1}^{L}(S_{i}^{z})^2,
$$
which has internal \(U(1)\rtimes Z_2\) symmetry, a \(\mathbb{Z}_2\times\mathbb{Z}_2\) subgroup generated by \(\pi\) rotations, and bond-centered inversion. For spin 1, the on-site \(\pi\)-rotation operators are
$$
U_\pi^\alpha = 1-2(S^\alpha)^2.
$$
The natural sector labels are \(\{M^z,X,Z,I\}\), where \(M^z\) is total magnetization and \(X,Z,I=\pm1\) are the eigenvalues of global \(U_\pi^x\), \(U_\pi^z\), and inversion [2311.14715].

The Kennedy–Tasaki unitary is
$$
U_{KT}=\prod_{1\le u<v\le L} e^{i\pi S^z_u S^x_v},
$$
with \((U_{KT})^2=1\) and \(U_{KT}^\dagger=U_{KT}\). It maps the original short-range Hamiltonian to another short-range Hamiltonian \(\tilde H=U_{KT}HU_{KT}\). In the dual model, the visible on-site symmetry is only \(\mathbb{Z}_2\times\mathbb{Z}_2\), while the original \(U(1)\) becomes a hidden non-local symmetry generated by
$$
Q_{\text{hidden}} = U_{KT}\left(\sum_i S_i^z\right)U_{KT}.
$$
The dual sectors are therefore labeled visibly by \(\{X,Z,I\}\), with an additional hidden \(U(1)\) 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 \(M^z\), is resolved. If one uses only the coarser \(\{Z,X,I\}\) labels, different \(M^z\) sectors are mixed and level repulsion is lost. In the dual model, visible \(\{Z,X,I\}\) sectors likewise show non-GOE statistics because the hidden \(U(1)\) charge remains unresolved. When perturbations such as \(D_x\sum_i (S_i^x)^2\), single-site defects \(D_x(S_1^x)^2\) or \(D_x(S_{L/2}^x)^2\), or a random field \(\sum_i h_i (S_i^x)^2\) 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 \((X,Z,I)\) and hidden \(U(1)\) quantum numbers that organize hidden order and determine the correct symmetry-sector decomposition.

## 6. SO\((n)\) 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\((n)\)-symmetric bilinear–biquadratic chain
$$
H = \sum_{j=1}^N \Big[ \cos\theta \sum_{a<b} T^{ab}_j T^{ab}_{j+1} + \sin\theta \Big(\sum_{a<b} T^{ab}_j T^{ab}_{j+1}\Big)^2 \Big].
$$
It has a Uimin–Lai–Sutherland point at
$$
\theta=\arctan\frac{1}{n-2},
$$
with low-energy limit \(\mathrm{SU}(n)_1\) WZW, a second integrable point at
$$
\theta=\arctan\frac{n-4}{(n-2)^2},
$$
and a special MPS point
$$
\theta=\arctan\frac{1}{n},
$$
where the ground states are exactly known SO\((n)\) AKLT states [2508.13114].

These SO\((n)\) AKLT states are built from Gamma matrices satisfying \(\{\Gamma^a,\Gamma^b\}=2\delta_{ab}\). The crucial structural datum is that the physical degrees of freedom transform in the vector representation of \(\mathrm{SO}(n)\), while the virtual MPS space carries a spinor representation of \(\mathrm{Spin}(n)\). For odd \(n\) there is a unique ground state; for even \(n\) 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 \(2^{\lfloor n/2\rfloor}\) as natural SPT-type index data.

At the continuum level, the work uses the conformal embedding \(\mathrm{Spin}(n)_2\subset \mathrm{SU}(n)_1\) to construct non-Cardy boundary states. For odd \(n\), the relevant topological defect line is \(\mathcal{D}_\sigma\), yielding a boundary state with
$$
g_{\mathcal{D}_\sigma}=n^{1/4}.
$$
For even \(n\), the relevant defects are \(\mathcal{D}_{\sigma_\pm}\), with
$$
g_{\mathcal{D}_{\sigma_\pm}}=\frac{n^{1/4}}{\sqrt{2}}.
$$
The overlaps of the SO\((n)\) AKLT MPS states with the \(\mathrm{SU}(n)\) ULS ground state have asymptotic form
$$
\ln|\langle\psi_0|\mathrm{MPS}\rangle|=-\alpha N+\ln g+\dots,
$$
and the exact calculation gives
$$
\ln g=\frac{1}{4}\ln n,\qquad g=n^{1/4}.
$$
The detailed discussion therefore presents the defect labels \(\mathcal{D}_\sigma\) or \(\mathcal{D}_{\sigma_\pm}\), the virtual spinor representation, and the Affleck–Ludwig boundary entropy \(g\) 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 \(\mathbb{Z}_2\) 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\((n)\) AKLT generalizations.

Source: https://www.emergentmind.com/topics/tasaki-index