67 2.15 (a) The energy spectrum is shown as a function of the chemical potential. b) The probability distribution of the energy-free mode as a function of site index j is shown. The upper half of the energy eigenvalue spectrum, superimposed with their corresponding IPRs, shows the extended, critical, and localized states. b) IPR associated with eigenstate indices as a function of λ for δ = 2.2 for a system of size L = 3194.
Anderson localization
Scaling theory of localization
The main idea of the scaling theory suggests that the conductivity of an ad-dimensional block of volume (BL)d (B is an integer) depends only on the conductivity of the Bd number of blocks corresponding to the volumeLd. The intersection acts as the critical transition point of the metal-insulator junction.
Optical lattice
QP potential in optical lattices
Although AL is observed in several systems, direct detection of the eigenstates in real experiments with materials makes it difficult to achieve. The speckle potential is random in nature, which is not the focus of this thesis.
Aubry-Andr´e model
Self-dual property
The localization transition in the AA model occurs because of the self-dualistic symmetry that is unique and inherent in the model. To get an accurate understanding of the self-duality in the AA model, we can follow the probability distribution of the eigenstate corresponding to the real and momentum space Hamiltonians Eq.
Harper model
In this study, it is observed that the eigenspectrum takes on a similar structure when plotted over a range of different rational values of β. Using this similar structure, Hofstadter was able to show that the spectrum for the irrational β formed the so-called Cantor group.
Characterization of localization properties
We numerically plot the average IPR and NPR as a function of the QP potential strength λ in Fig. We further show the probability distribution of the eigenstates as a function of the site index (j), corresponding to the extended and localized nature in the figure.
Experiment to theoretical model
To do this, the eigenstateψ is expanded in the Wannier basis of the lower band of the primary lattice, which can be written as, . Simplifying, we get, λ= Vos. 1.56) Therefore, the Hamiltonian of the bichromatic potential can be written as,.
Outline of the thesis
Multifractals analysis
Classification of intrinsic conditions such as localized, widespread or critical conditions can be done by calculating the inverse participation ratio (IPR). Therefore, the multifractal nature of the eigenstates can be recognized through the generalized IPR and its scaling exponent τq.
Eigenspectrum analysis
Level-spacing
2.7) Due to the presence of doubly degenerate eigenvalues in the extended phase, the value of se−on will be zero, while so−en will not be zero. However, a distribution of fluctuations for se-on and so-en will be present corresponding to the critical phase.
Hausdorff dimension
In this thesis, we will calculate another quantity, namely the energy level space, which uses eigenenergies. Suppose, the energy eigenvalues are arranged in ascending order, that is, E1 < E2 <. lt;ELwhereIs the length of the system.
Phase transition
Critical phenomena
The collective behavior of the critical point associated with the continuous phase transition known as critical phenomena. It has been observed that the size of droplets of CO2 becomes similar to the wavelength of the incident light when approaching the critical point from above.
Scaling theory
Therefore, a generalized homogeneous function will satisfy the following. 2.16) Similarly, the scaling hypothesis at the critical point can be expressed as,. Using these two exponents aandb, it is possible to find expressions between the critical exponents, known as scaling laws.
Critical state analysis
The critical potential strength λ and the exponent ratio γ/ν are determined from the abscissa and ordinate respectively for the common crossing point. In the vicinity of the critical point, a finite-size scaling form of the order parameter σ for finite system is defined by , .
Topology
Symmetries
Now we consider two situations that follow from the condition that |ψ⟩ remains invariant under the time-reversal operation. In the first situation, we apply the time-reversal operator (τ) on the state |ψ⟩ and then move forward in time byδt, which is given by,.
Classification of Topological Phases of Matter
Topological band theory
According to theadiabatic theorem, if we slowly change the Hamiltonian, in which case the system initially in its current state, will reach its instantaneous state of H(S⃗(t)) at the later time. The Berry phase is given by, . 2.56) Therefore, the Berry relation or Berry vector potential can be written as,.
Topological insulator: SSH model
Model
Next, the inter-cell hop strength between two subgrids of consecutive unit cells is given byt2.
Bulk properties
Furthermore, we study the properties of the vectord(k)⃗ in the dx−diplane within the BZ in Fig. Therefore, in addition to d(k)⃗ vector, the distribution relation performs the calculation of the winding number.
Finite-size properties
Thus, the zero-energy edge modes localized at the boundaries of the system characterize the topological properties of the model. According to this theory, topological invariants corresponding to bulk properties can be characterized by edge modes at system boundaries.
Topological superconductor: Kitaev chain
- Bulk properties
- Topological phase diagram
- Majorana zero modes
- Finite-size properties
- Real-space winding number
Similarly, each of the Majorana fermions can be written in terms of the Dirac fermions, given by, . 2.75], we will get the Hamiltonian in the Majorana basis. The topological (trivial) phase is characterized by the presence (absence) of the zero energy states.
Bose-Hubbard model
Phases of the BHM
As a result, the atoms tend to extend through the entire lattice, hence called the superfluid phase. All the atoms occupy the superfluid ground state corresponding to zero momentum (k= 0), which is given by [136],.
Phase diagram
Using the critical state analysis, discussed in the second chapter (sec. 2.3.3), we can validate the phase transition scenario and the critical point of the AA model [18]. 3.1 (b) we plot R[L,L′]as a function of λcorresponding to different sets of LandL′], which is shown in the legend of the figure.
Generalized AA model: case of modulated hopping
Phase diagram
In 3.5 we plot a phase diagram of the information obtained from the mean value of D2(⟨D2⟩) over all eigenstates as a function of λ1 and λ2. More attention is needed to understand the nature of the eigenstates belonging to the critical phase.
Finite-size scaling analysis
We thus plot the fractal dimension D2 for all eigenstates as a function of λ1 corresponding to λ2 = 0.0 in Fig. All eigenstates show a phase transition from an extended to a multifractal state at λ1 ∼ 1, resulting in a mobility advantage.
Conclusions
By calculating the fractal dimension, we found that the system hosts a phase transition from an extended phase to a multifractal phase in the absence of the on-site QP potential. This means that some of the localized states become conductive again for a range of the quasiperiodic potential.
Model
In particular, although the chiral symmetry of the SSH model is preserved in the case of finite hopping disorder, it is explicitly broken at any finite value of the strength of the disorder in place. Attempts have been made to understand such interesting scenarios in the context of the nature of chirality [144], the interplay between long-distance shopping and disorder [145], and also the possible existence of the mobility edge [146].
Results
- Uniform disorder
- Staggered disorder
- Phase diagram
- Edge modes
- Finite-size scaling
The presence of the critical region (blue region bounded by the symbols) is clearly distinguished from the fully extended or the fully localized regions (red regions) in the phase diagram. This re-entrant localization of the boundary conditions is slow as a function of λ in the case for largerδ.
Conclusions
In other words, for some specific dimerization strengths and as a function of the on-site potential, the system first demonstrates a localization transition where all individual particle states are localized. In this chapter we study the critical properties of the re-entry localization transition described above.
Model
While from our detailed analysis in this chapter, we are able to obtain the critical points, critical exponents, and scaling behavior associated with the first localization transition, which is the extended→intermediate→localized→intermediate transition, however , in the second localization transition, the scaling behavior is not well captured. Intercellular hopping between the two substrates is denoted met2 and het1 refers to hopping within the cell.
Results
- Phase diagram
- Multifractal analysis
- Critical state analysis
- Hausdorff dimension
Similar to the case of σ, for our analysis we use eigenstates corresponding to a narrow band of the spectrum to calculate the function R[L,L′]. The calculations were made by taking for the study the average of the states in the band with index/L=[0.45 to 0.5] of the energy spectrum.
Conclusions
The dimerized Kitaev chain in the presence of a random potential also showed a similar behavior [166]. Motivated by the above results, in this chapter we consider a dimerized Kitaev chain in the presence of on-site QP potential.
Model
To do this, we define a quasi-particle operator obtained by superposition of the single-particle creation (c†) and annihilation (c) operators, namely,. Using the BdG equations in (Eq. 6.5), the single-quasiparticle spectrum (En) and the amplitudesu(n)m,αandv(n)m,α of the wave function can be calculated.
Results
- Localization properties
- Phase diagrams
- Fractal dimension
- Level-spacing
- Probability distribution
- Finite-size analysis
- Topological properties
- Topological and universal Phase diagrams
- Bulk-boundary correspondence
- Probability distribution
- Finite-size analysis
On the other hand, the critical phase hosts a mixture of extended and localized states. Thus, we show some of the energy eigenvalues around zero energy as a function of λ corresponding to δ = 0.6 in Fig. 6.10 (a).
Conclusions
Here we show the presence of the topologically nontrivial phase via a shaded region that is in blue color. In addition, a non-trivial phase will undergo another transition to the Anderson localized phase at large values of the QP potential.
Model
By numerically calculating the site-decoupled Bose Hubbard Hamiltonian, we reconfirmed the presence of such phases. We further performed a finite size scaling analysis to determine the nature of the BG-SF or QM-SF phase transitions.
Results
- SF order parameter and compressibility
- Indicators of MI, BG, QM and SF phases
- Percolation appearance and cluster size distribution
- Finite-size scaling and critical exponents
- Phase Diagram
With gradual increase in potential strength, the BG phase now intrudes between the MI and SF phases by suppressing. However, the presence of the QM phase and thus determination of the BG-SF or QM-SF phase boundaries remains challenging.
Conclusions
Critical properties of the ground state localization-delocalization transition in the multiparticle aubry-andr´e model. Observation of slow dynamics near the many-body localization transition in one-dimensional quasiperiodic systems.
