Generalized criteria of symmetry breaking. A strategy for quantum time crystals

Abstract

The aim of this paper is to propose a criterion of spontaneous symmetry breaking that makes reference to the properties of pure phases defined by a translationally invariant state. By avoiding any reference to the ground state, at the basis of the standard approach, this criterion applies to a wider class of models. An interesting application is the breaking of time translations. Indeed, we discuss explicit theoretical models which exhibit the prototypical features of quantum time crystals, without the need of a time-dependent Hamiltonian.

Nonexistence of spontaneous symmetry breakdown of time-translation symmetry: a review

“…In fact, the derivation of Eq. ( 9) of [53], which leads to Equation (23), tacitly assumed translation invariance of these quantities. The recent preprint [38] reported that the lack of translation covariance of local order parameters can yield breakdown of (24) for a translation invariant vacuum state of some simple quantum field model.…”

“…Finite-range Hamiltonians are considered in [54]. The Lieb-Robinson bound estimate [32] for local Hamiltonians is applied to derive (23). It has been known that any finite-range quantum spin lattice model has an approximately inner {α t ∈ Aut(A), t ∈ R}, and that the Lieb-Robinson bound estimate yields an approximately inner time evolution, see [5] [39].…”

“…We agree that this limit describes temporal behavior of ρ β,Gib . However, regarding the general status of Griffiths-type LRO, "absence of quantum time crystals for all equilibrium states" cannot be concluded solely from (23) (24) without more elaborate reasoning on the model under consideration. Non-detection by a particular method is not proof of nonexistence of something, no matter how useful the method is to find its existence.…”

“…In the final part of [54], construction of quantum time crystals by a net of local states (densities) invariant under the local Heisenberg time evolutions (18) is considered. In [23], another method of generating quantum time crystals by stationary states is provided with a concrete example of quantum spin lattice models. In C * -algebraic language, a general framework for such quantum time crystals can be formulated as follows.…”

Corrections to Wigner–Eckart Relations by Spontaneous Symmetry Breaking

“…With the advent of spontaneous symmetry breaking, a mechanism typically appearing in infinitely extended systems, or in any case for systems admitting more than one inequivalent realization (see e.g. [2,3]), the Wigner-Eckart strategy appears to be completely out of use; first because the symmetries of the Hamiltonian may be completely lost at the level of the states. Second, only in special cases the effect of spontaneous symmetry breaking may be considered small and treated by means of a perturbative expansion.…”