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Tesi etd-12012015-092306


Thesis type
Tesi di dottorato di ricerca
Author
NESPOLO, JACOPO
URN
etd-12012015-092306
Title
Scaling behaviour of quantum systems at thermal and quantum phase transitions
Settore scientifico disciplinare
FIS/02
Corso di studi
FISICA
Commissione
tutor Prof. Vicari, Ettore
Parole chiave
  • cold atoms
  • inhomogeneous quantum systems
  • critical phenomena
Data inizio appello
26/12/2015;
Consultabilità
completa
Riassunto analitico
Experimental setups are finite in space and hardly ever in homogeneous <br>conditions.<br>This is very different from the ideal settings of the thermodynamic limit <br>often adopted in condensed matter theories.<br>Therefore, close to phase transitions, where typically long range correlations<br>build up, it is important to correctly take into account the way in which<br>boundaries and inhomogeneities affect the critical behaviour.<br>This can be achieved by means of the finite-size (FSS) and trap-size (TSS)<br>scaling theories, which generally apply to continuous phase transitions, where<br>one can define a diverging length scale.<br>FSS and TSS are reviewed in the first part of this work, together with some<br>general properties of systems close to phase transitions.<br><br>We then numerically study the TSS properties of the continuous<br>finite-temperature phase transition of the Bose-Hubbard model (BH) in two and<br>three dimension.<br>This quantum model realistically describes experiments with ultra-cold bosonic<br>gases trapped in optical lattices.<br>In three dimensions, the BH exhibits a standard normal-to-superfluid<br>transition.<br>In two dimensions, the transition becomes of the Berezinski-Kosterlitz-Thouless<br>type, characterised by logarithmic corrections to scaling.<br>We perform thorough FSS analyses of quantum Monte Carlo data in homogeneous<br>conditions to extract the value critical temperature.<br>In two dimensions, this requires devising a matching method in which the FSS<br>behaviour of the 2D BH is matched to the classical 2D XY model, whose<br>transition belongs to the same universality class.<br>We subsequently verify the validity of the TSS ansatz by simulating the trapped<br>systems at the critical temperature.<br>We find that the TSS theory is general and universal once one takes into<br>account the effective way in which the trapping potential couples to the<br>critical modes of the system.<br><br>In the last part of this Thesis, we extend the FSS and TSS to discontinuous (or<br>first order) quantum phase trnasitions.<br>Discontinuous transitions do not develop a diverging length scale in the<br>thermodynamic limit, but are rather characterised by the coexistence of<br>domains of different phases at the transition.<br>The typical size of single-phase domains induce a behaviour that closely<br>resembles finite size scaling.<br>We find that the scaling variable that parametrises the scaling behaviour at<br>discontinuous transitions is the ratio of the perturbation energy driving the<br>transition to the finite-size energy gap.<br>We further find that inhomogeneous systems exibiting first order transitions<br>can be treated heuristically in analogy with the TSS behaviour at continuous<br>transitions.<br>These findings are confirmed numerically on the quantum Ising and quantum Potts<br>chains, which are simulated using density matrix renormalisation group<br>techniques.
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