ETD

Archivio digitale delle tesi discusse presso l'Università di Pisa

Tesi etd-07032017-183808


Tipo di tesi
Tesi di laurea magistrale
Autore
SURACE, FEDERICA MARIA
URN
etd-07032017-183808
Titolo
Floquet time crystals in clock models
Dipartimento
FISICA
Corso di studi
FISICA
Relatori
relatore Prof. Fazio, Rosario
Parole chiave
  • time crystal
  • clock model
Data inizio appello
20/07/2017
Consultabilità
Completa
Riassunto

Spontaneous symmetry breaking is a concept of fundamental relevance in modern theoretical physics. Realizations of a physical system that differ for their symmetry represent different states of matter: a symmetry breaking is related to a change in the phase of the system. One of the examples we are most familiar with is the phase transition from liquid to crystal that is associated to the spontaneous breaking of spatial translation symmetry.

In analogy with crystals in space, Wilczek recently proposed that spontaneous breaking of time translation symmetry can produce a ``time crystal``. This idea immediately raised both enthusiasm and criticism. Few years later, a no go theorem was proven, showing that time crystals cannot exist at thermal equilibrium.
However, the theorem does not exclude time translation symmetry breaking in non-equilibrium settings. In fact, periodically driven many body systems can exhibit time crystalline behaviour, as has already been demonstrated in experiments.
Systems with a periodic drive have an underlying discrete time translation symmetry that can be spontaneously broken in a less symmetric phase. If the Hamiltonian is periodic with a period $T$, in the time crystal phase the system will respond with a period $nT$.
These subharmonic oscillations of physical observables associated to time translation symmetry breaking have some characteristic features that are not found in other similar phenomena occuring in non-correlated systems. In fact, time crystals originate from a many-body effect that crucially relies on the presence of interactions.
As a consequence of this collective nature, oscillations with a multiple of the original period are remarkably stable to perturbations and persist for an infinitely long time in the thermodynamic limit.

The first model that was proposed as an example of discrete time crystal consisted of a disordered spin chain with Ising interactions undergoing a periodic spin flip pulse. Disorder plays an essential role because it prevents the chain from absorbing energy indefinitely from the drive and to heat to an infinite temperature state.
The model exhibited the emergence of a phase with robust period doubling. Different experiments were performed that confirmed the predicted time crystalline oscillations: one has been realized using trapped atomic ions interacting via optical dipole forces; another one used an ensemble of nitrogen vacancy spin impurities in diamond.

The main purpose of this work is to examine both analytically (where possible) and with numerical simulations the existence of a time crystal phase in a more general class of models. The paradigmatic example of a discrete time crystal with period doubling is based on an Ising chain, where a $\mathbb{Z}_2$ symmetry is spontaneously broken. Therefore, an analogous model with an underlying $\mathbb{Z}_n$ symmetry (the so called n-state clock model) and an appropriate drive of period $T$, is expected to result in a time crystal with period $nT$.

In this thesis, the case of a 3-state clock model is first considered. Analytic calculations for an exactly solvable case show that two different situations are possible: if the model is non-chiral, i.e. if the parameters in the Hamiltonian are such that the interactions have an additional time reversal symmetry, a time crystal is not possible due to the degeneracies in the spectrum; on the other hand, if the model is chiral (with no time reversal symmetry) the system is expected to be a time crystal. This result can be generalized when a perturbation is added and the model is no longer exactly solvable. Assuming the existence of a ''local spectral gap``, it can be deduced that the time crystal phase is not destroyed provided that the perturbation is sufficiently small.

This prediction for the 3-state clock model is tested via exact diagonalization of finite size systems. Useful quantities are identified in order to analyze the persistence of the $3T$-periodic oscillations and the spectral properties of the system. The scaling of these quantities with the system size confirms the results predicted in the thermodynamic limit.

The general case of the n-state clock model is also discussed. A time crystal with period $nT$ is expected as long as the spectrum has no local degeneracies: the corresponding condition on the parameters of the Hamiltonian is derived. In conclusion, a slightly more general model is considered, showing that the n-state clock model can generate different time crystal phases depending on the drive.
File