In this thesis, we have explored the commonalities and connections between different classes of quantum systems that do not thermalize. Specifically, we have (1) shown that localized systems possess conservation laws like integrable systems, which can be constructed in a systematic way and used to detect localization-delocalization transitions
, (2) studied the phenomenon of many-body localization in a model with a single
particle mobility edge, (3) shown that interesting finite-size scaling emerges, with universal exponents, when athermal quantum systems are forced to thermalize through the
application of perturbations and (4) shown that these scaling laws also arise when a perturbation causes a crossover between quantum systems described by different random
matrix ensembles. We conclude with a brief summary of each chapter.
In Chapter 2, we have investigated the effects of finite size on the crossover between quantum integrable systems and non-integrable systems. Using exact diagonalization of finite-sized systems, we have studied this crossover by obtaining the energy level statistics and Drude weight associated with transport. Our results reinforce the idea that for system size L ā ā, non-integrability sets in for an arbitrarily small integrabilitybreaking
perturbation. The crossover value of the perturbation scales as a power law
ā¼ Lā3 when the integrable system is gapless and the scaling appears to be robust to
microscopic details and the precise form of the perturbation.
In Chapter 3, we have studied the crossover among different random matrix ensembles
CHAPTER 6. CONCLUSION 127
[Poissonian, Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE)
and Gaussian Symplectic Ensemble (GSE)] realized in different microscopic models. We
have found that the perturbation causing the crossover among the different ensembles
scales to zero with system size as a power law with an exponent that depends on the
ensembles between which the crossover takes place. This exponent is independent of
microscopic details of the perturbation. We have also found that the crossover from the
Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover
which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE
associated with symmetry breaking introduces a subdominant contribution. Finally,we
have conjectured that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the
dimensionality of the system.
In Chapter 4, we have outlined a procedure to construct conservation laws for Anderson
localized systems. These conservation laws are found as power series in the hopping
parameters. We have also obtained the conservation laws for the disorder free Aubry-Andre model, where the states are either localized or extended depending on the strength of a coupling constant. We have formulated a specific procedure for averaging over disorder, in order to examine the convergence of the power series. Using this procedure for the Aubry-Andre model, we show that integrals of motion given by our construction are well-defined in the localized phase but not so in the extended phase. Finally, we also obtain the integrals of motion for a model with interactions to lowest order in the interaction.
In Chapter 5, we have studied many body localization and investigated its nature
in the presence of a single particle mobility edge. Employing the technique of exact diagonalization for finite-sized systems, we have calculated the level spacing distribution, time evolution of entanglement entropy, optical conductivity and return probability to characterize the nature of localization. The localization that develops in the presence of interactions in these systems appears to be different from regular Many-Body Localization (MBL) in that the growth of entanglement entropy with time is linear (like in
CHAPTER 6. CONCLUSION 128
a thermal phase) instead of logarithmic but saturates to a value much smaller than the
thermal value (like for MBL). All other diagnostics seem consistent with regular MBL
| Identifer | oai:union.ndltd.org:IISc/oai:etd.iisc.ernet.in:2005/3888 |
| Date | January 2015 |
| Creators | Ranjan Krishna, M |
| Contributors | Mukherjee, Subroto |
| Source Sets | India Institute of Science |
| Language | en_US |
| Detected Language | English |
| Type | Thesis |
| Relation | G27289 |
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