# Systematic Construction of Counterexamples to the Eigenstate Thermalization Hypothesis

## TL;DR

This paper introduces a method to embed specific states into the energy spectrum of many-body Hamiltonians, creating models that violate the eigenstate thermalization hypothesis while still exhibiting thermalization behavior.

## Contribution

A novel construction technique for Hamiltonians with targeted eigenstates that violate ETH, providing analytical proofs and numerical demonstrations.

## Key findings

- Constructed Hamiltonians with no local conserved quantities that violate ETH.
- Target states are non-thermal, while other eigenstates thermalize.
- Models thermalize after a quench despite ETH violation.

## Abstract

We propose a general method to embed target states into the middle of the energy spectrum of a many-body Hamiltonian as its energy eigenstates. Employing this method, we construct a translationally-invariant local Hamiltonian with no local conserved quantities, which does not satisfy the eigenstate thermalization hypothesis. The absence of eigenstate thermalization for target states is analytically proved and numerically demonstrated. In addition, numerical calculations of two concrete models also show that all the energy eigenstates except for the target states have the property of eigenstate thermalization, from which we argue that our models thermalize after a quench even though they does not satisfy the eigenstate thermalization hypothesis.