Symmetric Trivial Gapped Ground States

• Symmetric trivial gapped ground states are unentangled product states that are invariant under global Hamiltonian symmetries and exhibit a strictly positive energy gap.
• They are realized in frustration-free spin chains through rank-1 projectors that enforce zero ground state energy and facilitate mapping between different symmetry sectors.
• Emergent non-invertible symmetries in these models lead to spontaneous symmetry breaking, with distinct order parameters differentiating inequivalent ground state representations.

A symmetric trivial gapped ground state is defined as a ground state of a quantum many-body system that (i) is invariant under the global symmetries of the Hamiltonian, (ii) is gapped (i.e., the energy to the first excited state is strictly positive in the thermodynamic limit), and (iii) is trivial in the sense of being short-range entangled—often realized as a product state or a state connected by a finite-depth local unitary circuit to a product state. The structure, classification, and realization of such states are central topics in the theory of quantum phases of matter, especially in the context of frustration-free spin chains, symmetry-protected topological phases, and the role of (non-)invertible symmetries.

1. Frustration-Free Spin Chains and Trivial Ground States

In frustration-free spin chains, the Hamiltonian is constructed such that the global ground state minimizes every local term: $$H_{af} = \sum_{j=1}^{N} h_{j,j+1} = \sum_{j=1}^{N} A_j F_{j+1}$$ where $A$ and $F$ are Hermitian projectors. In the rank-1 case, these are explicitly

$$A = \frac{1}{1+a^2}\begin{pmatrix}1 & a \ a & a^2\end{pmatrix}, \quad F = \frac{1}{1+f^2}\begin{pmatrix}1 & f \ f & f^2\end{pmatrix}$$

with $a, f \in \mathbb{R}$. The frustration-free property implies that the ground state energy is exactly zero and the ground states can be determined by finding the simultaneous null spaces of all projectors.

The two product ground states can be written as

$$|g_{a}\rangle = \bigotimes_j |0_a\rangle, \quad |g_{f}\rangle = \bigotimes_j |0_f\rangle$$

with

$$|0_x\rangle = \frac{1}{\sqrt{1+x^2}}\begin{pmatrix} -x \ 1 \end{pmatrix} \quad (x = a, f)$$

These states are unentangled, symmetric under local operations, and trivial in the sense of being continuously deformable from product states as long as symmetry is either ignored or not strong enough to distinguish distinct classes.

2. Gap Criteria and Spectral Properties

The existence of a gap above the ground state is crucial for the stability and triviality of the phase. The gap can be analyzed via the spectral properties of an auxiliary matrix $T_p$, constructed from overlaps of the states associated with $A$ and $F$. The specific structure is: $$T_p = \begin{pmatrix} -\langle1_a,1_f|0,1\rangle & \langle1_a,1_f|1,1\rangle \ -\langle1_a,1_f|0,0\rangle & \langle1_a,1_f|1,0\rangle \end{pmatrix}$$ with $|1_x\rangle = \frac{1}{\sqrt{1+x^2}}(1, x)^T$. The eigenvalues are $\{0, \frac{f-a}{(1+a^2)(1+f^2)}\}$. As long as $a \neq f$, one eigenvalue is nonzero, ensuring a strictly positive gap. This meets the requirements of the Bravyi-Gosset criterion for spectral gappiness in frustration-free systems (Sinha et al., 17 Feb 2025).

The presence of a gap ensures that the system does not host low-lying excited states that could destabilize the trivial phase or promote additional phase structure.

3. Non-Invertible Symmetries and Spontaneous Symmetry Breaking

Despite the triviality (product structure) of the ground states, the system possesses global non-invertible symmetries. These are not generated by invertible group actions or dualities; instead, they are implemented via global operators constructed from the local complementary projectors: $$S_f = \prod_j F_j^\perp, \qquad S_a = \prod_j A_j^\perp$$ where $F_j^\perp$, $A_j^\perp$ project onto the orthogonal complement of the ground state sector for $F$ and $A$ respectively.

The symmetry operators map between the two ground states: $$S_f |g_a\rangle = \left( \frac{1+af}{\sqrt{1+f^2}\sqrt{1+a^2}} \right)^N |g_f\rangle, \quad S_a |g_f\rangle = \left( \frac{1+af}{\sqrt{1+f^2}\sqrt{1+a^2}} \right)^N |g_a\rangle$$ while $S_f |g_f\rangle = |g_f\rangle$ and $S_a |g_a\rangle = |g_a\rangle$.

In the thermodynamic limit ($N\to\infty$), the prefactor becomes vanishingly small unless $a=-f$; thus, transitions between $|g_a\rangle$ and $|g_f\rangle$ are suppressed, resulting in spontaneous breaking of the non-invertible symmetry—each ground state is invariant under only one symmetry operator. The symmetry does not arise from a duality and cannot be interpreted as a group symmetry.

4. Algebraic Quantum Theory and Order Parameters

The notion of symmetry breaking is formalized using the tools of algebraic quantum theory. The observable algebra for the infinite chain is the quasi-local $C^*$-algebra

$$\mathcal{B} = \bigotimes_{i \in \mathbb{Z}} \mathrm{Mat}_2(\mathbb{C})$$

Ground states $|g_a\rangle$, $|g_f\rangle$ generate Hilbert spaces $\mathcal{H}_a$, $\mathcal{H}_f$ carrying inequivalent representations $\pi_a$, $\pi_f$ of $\mathcal{B}$.

A non-invertible symmetry-breaking order parameter is constructed as

$$\mathcal{O} = \lim_{N\to\infty} \frac{1}{N} \sum_{j=1}^N (F + FA - A - AF)_j$$

Expectation values distinguish the sectors: $$\pi_a(\mathcal{O}) = \sigma \, \mathrm{Id}, \quad \pi_f(\mathcal{O}) = -\sigma \, \mathrm{Id}$$ with $\sigma = \frac{(a-f)^2}{(1+a^2)(1+f^2)}$. The lack of any intertwiner between $\pi_a$ and $\pi_f$ realizes algebraically the spontaneous breaking of the non-invertible symmetry: the two ground states define superselection sectors.

5. Trivial Gapped Ground States: Classification and Role of Projectors

The rank-1 character of projectors $A$ and $F$ is central. The ground states are annihilated by the projectors, resulting in product states. The non-invertible symmetry operators are constructed from the complementary projectors, which both ensure symmetry of the Hamiltonian and enable the symmetry operations that map between (but cannot mix) the two product ground states in the thermodynamic limit.

This explicit construction shows that even in the simplest possible gapped phase—composed of unentangled, symmetric product ground states—non-invertible symmetries can exist and be spontaneously broken. The rank of the projectors determines the local constraint structure and, by extension, the structure of the non-invertible symmetry as well as the possibility for additional ground state degeneracies if higher-rank projectors are used.

6. Context and Implications

This realization of symmetric, trivial, gapped ground states with spontaneous breaking of non-invertible symmetries enriches the landscape of possible symmetry concepts in quantum spin chains. Unlike SPT phases, where nontrivial entanglement structure is protected by (usually invertible) group symmetries, here the ground states are trivial in their entanglement but support an emergent non-invertible symmetry structure that is not tied to dualities or higher categories and is proven using the rigorous framework of algebraic quantum theory (Sinha et al., 17 Feb 2025).

Such models underscore that spontaneous symmetry breaking—hitherto mostly associated with invertible, group-like symmetries—can manifest for non-invertible symmetries even in fully frustration-free, product-like gapped phases. This challenges the intuition that only phases with intricate entanglement can display interesting symmetry phenomena and suggests potential further generalizations to higher-dimensional and nontrivial projector or MPO-based symmetries.

7. Summary Table: Key Model Elements

Feature	Construction / Outcome	Mathematical Expression
Hamiltonian	Frustration-free sum of projectors	$H_{af} = \sum_j A_j F_{j+1}$
Ground states	Product (ferromagnetic) states	$|g_a\rangle$, $|g_f\rangle$
Non-invertible symmetries	Global operators from projector complements	$S_f$, $S_a$
Gapped nature	Auxiliary matrix $T_p$ eigenvalues	$\{0, \frac{f-a}{(1+a^2)(1+f^2)}\}$
Symmetry breaking	Distinct order parameters in representations	$\pi_a(\mathcal{O}) = \pm \sigma \Id$

This synthesis demonstrates how exact solvability, symmetry, and spontaneous symmetry breaking interact even in minimal (trivial) gapped systems, providing rigorous platforms for further exploration of non-invertible symmetry phenomena.