$W$ state is not the unique ground state of any local Hamiltonian

TL;DR

This paper demonstrates that the $W$ state cannot be the unique ground state of any local Hamiltonian, revealing its inherent degeneracy and implications for quantum phase classification and stability analysis.

Contribution

It introduces a new class of states, including the $W$ state, that can only appear as degenerate ground states, providing insights into their stability and role in quantum phase transitions.

Findings

$W$ state cannot be a unique ground state of local Hamiltonians.

States in this class are always degenerate, even in gapless or disordered models.

Degeneracy can serve as a signature of certain quantum critical points.

Abstract

The characterization of ground states among all quantum states is an important problem in quantum many-body physics. For example, the celebrated entanglement area law for gapped Hamiltonians has allowed for efficient simulation of 1d and some 2d quantum systems using matrix product states. Among ground states, some types, such as cat states (like the GHZ state) or topologically ordered states, can only appear alongside their degenerate partners, as is understood from the theory of spontaneous symmetry breaking. In this work, we introduce a new class of simple states, including the $W$ state, that can only occur as a ground state alongside an exactly degenerate partner, even in gapless or disordered models. We show that these states are never an element of a stable gapped ground state manifold, which may provide a new method to discard a wide range of 'unstable' entanglement area law states in the numerical search of gapped phases. On the other hand when these degenerate states are the ground states of gapless systems they possess an excitation spectrum with $O(1/L^2)$ finite-size splitting. One familiar situation where this special kind of gaplessness occurs is at a Lifshitz transition due to a zero mode; a potential quantum state signature of such a critical point. We explore pathological parent Hamiltonians, and discuss generalizations to higher dimensions, other related states, and implications for understanding thermodynamic limits of many-body quantum systems.