97 research outputs found
Topology and Quantum Phases of Low Dimensional Fermionic Systems
No full textIn this thesis, we study quantum phase transitions and topological phases in low dimensional fermionic systems. In the first part, we study quantum phase transitions and the nature of currents in one-dimensional systems, using eld theoretic techniques like bosonization and renormalization group. This involves the study of currents in Luttinger liquids, and the fate of a persistent current in a 1D system. In the second part of the thesis, we study the different types of Majorana edge modes in a 1D p-wave topological superconductor. Further we extend our analysis to the e ect of an additional s-wave pairing and a Zeeman field on the topological properties, and present a detailed phase diagram and symmetry classification for each of the cases. In the third part, we concentrate on the topological phases in two-dimensional systems. More specifically, we study the experimental realization of SU(3) topological phases in optical lattice experiments, which is characterized by the presence of gapless edge modes at the boundaries of the system. We discuss the specific characteristics required by a such a three component Hamiltonian to have a non-zero Chern number, and discuss a schematic lattice model for a possible experimental realization.
The thesis is divided into three chapters, as discussed below:
In the first chapter, we study the effect of a boost (Fermi sea displaced by a finite momentum) on one dimensional systems of lattice fermions with short-ranged interactions. In the absence of a boost such systems with attractive interactions possess algebraic superconducting order. Motivated by physics in higher dimensions, one might naively expect a boost to weaken and ultimately destroy superconductivity. However, we show that for one dimensional systems the e ect of the boost can be to strengthen the algebraic superconducting order by making correlation functions fall o more slowly with distance. This phenomenon can manifest in interesting ways, for example, a boost can produce a Luther-Emery phase in a system with both charge and spin gaps by engendering the destruction of the former.
In the second chapter, we study the type of Majorana modes and the topological phases that can appear in a one-dimensional spinless p-wave superconductor. We have considered two types of p-wave pairing, 4"" = 4## and 4"" = 4##., and show that in both cases two types of Majorana bound states (MBS) with different spatial dependence emerge at the edges: one purely decaying and one damped oscillatory. Even in the presence of a Zeeman term B, this nature of the MBS persists in each case, where the value of chemical potential and magnetic field B decides which type will appear. We present a corresponding phase diagram, indicating the number and type of MBS in the -B space. Further, we identify the possible symmetry classes for the two cases (based on the ten-fold classification), and also in the presence of perturbations like a s-wave pairing and various terms involving magnetic field. It is seen that in the presence of a s-wave perturbation, the MBS will now have only one particular nature, the damped oscillating behaviour, unlike that for the unperturbed p-wave case.
In the third chapter, we study SU(3) topological phases in two dimension. It is shown by Barnett et.al that N copies of the Hofstadter model with 2N Abelian ux per plaquette is equivalent to an N-component atom coupled to a homogeneous non-Abelian SU(N) gauge field in a square lattice. Such models have non-zero Chern number and for N = 3, can be written in terms of the SU(3) generators. In our work, we uncover two salient ingredients required to express a general three-component lattice Hamiltonian in a SU(3) format with non-trivial topological invariant. We nd that all three components must be coupled via a gauge eld, with opposite Bloch phase (in momentum space, if the NN hopping between two components is teik, then for the other two components, this should be te ik) between any two components, and there must be band inversion between all three components in a given eigenstate. For spinless particles, we show that such states can be obtained in a tripartite lattice with three inequivalent lattice sites, in which the Bloch phase associated with the nearest neighbor hopping acts as k-space gauge eld. The second criterion is the hopping amplitude t should have an opposite sign in the diagonal element for one of the two components, which can be introduced via a constant phase ei along the direction of hopping. The third and a more crucial criterion is that there must also be an odd-parity Zeeman-like term (as k ! k, the term changes sign), i.e. sin(k) z term, where z is the third Pauli matrix defined with any two components of the three component basis. In the presence of a constant vector potential, the kinetic energy of the electron gets modified when the vector potential causes a flux to be enclosed. This can generate the desired odd parity Zeeman term, via a site-selective polarization of the vector potential. This can be achieved in principle by suitable modifications of techniques used in Sisyphus cooling, and with a suitable arrangement of polarizer plates, etc. The topological phase is a firmed by edge state calculation, obeying the bulk-boundary correspondence
Magnetisation, Phases & Phase Transitions in Frustrated and Unfrustrated XY Model
No full textThrough our whole work we study the XY model with all its entirety, a particular spin model where spins are confined in a plane. We try to bring out a good understanding of this model with all different types of phases and phase transition, it undergoes in critical situations. We conceive of these external conditions from very different physical models like High Tc Superconductor, Ultracold atoms in optical lattice which are in focus of recent research. Firstly we model high Tc Superconductors with very simple 2D XY model to get an idea about the diamagnetic response exhibited by these materials when kept in a external magnetic field. This modeling is reasonable because most of the physics of cuprate High Tc Superconductors are governed by their 2D copper oxide planes which insists us to consider 2D models. Later we shifted to a more realistic 3D anisotropic XY model , as the coupling between cuprates plane may have a considerable role in devising physics of those materials. We particularly focus on the 2D to 3D crossover effect on magnetisation showed by these models, with keeping an eye on how all these can be relate to the experimentally acquired magnetisation profile of High Tc Supercondutors. On the second project we investigate on the phase diagram of a fully frustrated 2-leg ladder Bose Hubbard model. After mapping it properly to a classical model, a bi-layer Fully Frustrated XY model on square lattice, we found that the frustration leads to the emergence of a new phase "Chiral Mott insulator(CMI)" sandwiched between "Chiral Superfluid(CSF)" and "regular Mott insulator(MI)" phase.
We divide the whole report into four parts. The first chapter is basically contain-ing introductory part comprising the motivation. In the second chapter we discuss various types of phases and phase transitions of the 2D & 3D XY models. We try to address their critical behaviors. In the third chapter and onwards we consider our model in external magnetic field and observe magnetisation in these systems. Here we specially focus on 2D to 3D crossover effect on magtisation measurement. Lastly in the fourth chapter we bring out a correspondence of XY model with the 2 leg ladder fully frustrated Bose Hubbard Model. There we report the emergence of a new phase, Chiral Mott Insulator(CMI) due to frustration in system
Thermalization and its Relation to Localization, Conservation Laws and Integrability in Quantum Systems
No full textIn 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 → ∞, non-integrability sets in for an arbitrarily small integrabilitybreaking
perturbation. The crossover value of the perturbation scales as a power law
∼ L−3 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 MB
Non-ergodicity in Interacting Quasiperiodic Systems and Disordered Fock Lattice Models
No full textIn this thesis, we study different aspects of many body localization in closed quantum systems. Many body localized systems are isolated interacting quantum systems that fail to thermalize on their own (non-ergodic) and thus violates the eigenstate thermalization hypothesis (ETH), which guarantees thermalization in quantum systems. These systems are known to have area law entanglement entropy even at high energy densities and have been argued to have emergent conservation laws, which prevent thermalization. The presence of thermal-MBL transitions has been confirmed in the interacting disordered systems and in the presence of deterministic quasiperiodic potentials, at least in one dimension. Initially, the MBL phase was introduced by showing that localization persists even in the presence of interactions in a system with a localized single particle spectrum. However, the fate of many body localization in interacting systems with coexisting localized and extended single particle states has been questioned recently and has been shown to be model-dependent. In the first work, we propose a dimensionless criterion based on the single particle spectrum, which can determine the presence or absence of the thermal-MBL transitions in the interacting quasiperiodic systems with coexisting localized and extended states. In the second work, we calculate the transport properties and the level spacing statistics in an interacting one dimensional system in the presence of similar quasiperiodic potential. The many-body spectrum of such a quasiperiodic system has been argued to have a non-ergodic extended phase, which is associated with the violation of ETH and volume law satisfying entanglement entropy. In this work, we show sub-diffusive transport in this non-ergodic extended phase in contrast to the diffusive transport in the thermal phase and no transport in the MBL phase. In the third work, we consider a tight-binding model in the Fock lattice with correlated onsite disorders and show the presence of a localization transition in such Fock lattice models. We consider different functional forms of correlations for the onsite Fock lattice potentials keeping the effective disorder strength fixed and discuss the possibility of localization transitions driven by the correlation among the onsite terms. In the fourth work, we develop a recursive method to calculate the exact Green’s functions in the Fock lattice, where each slice of the Fock lattice is added in recursion while calculating different elements of the total Green’s function. Using this method, we calculate different quantities to locate the thermal-MBL transition in interacting disordered systems as an alternative to the exact diagonalization method typically used in studies of MBL systems
A Theoretical Study of Thermoelectric Effect for Cross-plane Transport Across Twisted Bilayer Graphene
No full textThe thermoelectric effect and the linear response theory is one of the most studied areas in condensed matter physics. In the last few years, the study of Van Der Waals heterostructure has become one of the most promising domains of research in physics and nanoscience both theoretically and experimentally. Twisted bilayer graphene is synthesized by stacking one graphene layer on top of another and introducing a relative twist between them.
In this project, we have looked at the cross-plane electrical and thermal transport and the thermoelectric effect across twisted bilayer graphene. We have tried to give a theoretical framework based on the experiments performed in Prof. Arindam Ghosh's lab. In the entire project, we only looked at the thermoelectric transport for a large incommensurate twist angle (>10 degree). For these twist angles, the entire structure does not form any Moire superlattice and the two graphene layers become effectively decoupled.
The cross-plane transport across TBG is essentially ballistic so we have constructed a phenomenological framework using the Landauer formula to determine the current. In TBG, the inter-planer transport can take place via two processes, (1) The incoherent electronic tunneling which follows the Landauer formalism with a given transmission coefficient T(E) and (2) The phonon drag effect where the electrons are kicked out form a graphene plane by the thermally excited phonons having polarization mode perpendicular to the graphene plane. Experimentally it was found that at low temperature, the thermopower is governed by the incoherent electronic transmission process whereas, at high temperature, the phonon drag effect determines the form of Seebeck Coefficient. In this project, we have discussed the incoherent electronic transmission part both qualitatively and quantitatively. Here the cross-plane transport is considered as a perfect (zero scattering) ballistic transport and hence the transmission coefficient T(E) is taken as 1.This idea came from a purely phenomenological point of view which was first discussed in this paper arXiv:1801.01269. We have shown that at the low-temperature regime, the thermopower saturates to the Mott value. For thermal transport, the Wiedemann-Franz law is holding at the low-temperature regime. In the high-temperature regime, the proportionality constant between the ratio of thermal to electrical conductivity and absolute temperature of the system saturates to a finite value. In the end, we have also calculated the thermoelectric figure of merit (the ZT factor) for the inter-planner electronic transport across TBG and derived the efficiency as a function of absolute temperature
Studies of Diamagnetism and Thermoelectric Transport in High Temperature Superconductors and Graphene
No full textUsing a recently proposed Ginzburg-Landau-like free energy functional due to Banerjee et al. Phys. Rev. B 83, 024510 (2011) we calculate the fluctuation diamagnetism of high-T c superconductors as a function of doping, magnetic field, and temperature. We analyze the pairing fluctuations above the superconducting transition temperature in the cuprates, ranging from the strong phase fluctuation dominated underdoped limit to the more conventional amplitude fluctuation dominated overdoped regime. We show that a model where the pairing scale increases and the superfluid density decreases with underdoping produces features of the observed magnetization in the pseudogap region, in good qualitative and reasonable quantitative agreement with the experimental data. In particular, we explicitly show that even when the pseudogap has a pairing origin the magnetization actually tracks the superconducting dome instead of the pseudogap temperature, as seen in experiment. We discuss the doping dependence of the ‘onset’ temperature for fluctuation diamagnetism and comment on the role of vortex core-energy in our model.
We also study the Nernst effect in fluctuating superconductors by calculating the transport coefficient α_xy in a phenomenological model where the relative importance of phase and amplitude fluctuations of the order parameter is tuned continuously to smoothly evolve from an effective XY model to more conventional Ginzburg-Landau description. To connect with a concrete experimental realization we choose the model parameters appropriate for cuprate superconductors and calculate α_xy and the magnetization M over the entire range of experimentally accessible values of the field, temperature, and doping. We argue that α_xy and M are both determined by the
equilibrium properties of the superconducting fluctuations (and not their dynamics) despite the former being a transport quantity. Thus, the experimentally observed correlation between the Nernst signal and the magnetization arises primarily from the correlation between α_xy and M. Further, there exists a dimensionless ratio M/(T α_xy ) that quantifies this correlation. We calculate, for the first time, this ratio over the entire phase diagram of the cuprates and find it agrees with previous results obtained in specific parts of the phase diagram. We conclude that there appears to be no sharp distinction between the regimes dominated by phase fluctuations and Gaussian fluctuations for this ratio in contrast to α_xy and M individually. The utility of this ratio is that it can be used to determine the extent to which superconducting fluctuations contribute to the Nernst effect in different parts of the phase diagram given the measured values of magnetization.
In the fourth chapter, we study the thermoelectric transport properties across twisted bilayer graphene. In twisted bilayer graphene, two individual graphene layers are placed within van der Waals separation with a relative twist angle between them. The charge and heat transport of such a system has been a recent focus of research from the perspective of fundamental physics and possible applications in nanoscale devices. To understand the thermoelectric transport properties of a recent experiment which measures the Seebeck effect across twisted bilayer graphene, we develop a phenomenological model based on the Landauer-Büttiker transport formalism. We analyze the measurement of the Seebeck coefficient with a calculation that takes into account the tunneling properties of Dirac electrons through a barrier as well as incorporates a disorder potential giving rise the electron-hole puddles near the charge neutral point. By performing a detailed analysis it appears that the measured thermopower is determined by the cross-plane layer-breathing mode rather than the properties of the tunnel junction
An experimental study on thermoelectric transport at van der Waals interfaces
No full textWhen two planar atomic or molecular layers are brought to sub-nanometer proximity,
they form a van der Waals interface because the van der Waals force forms the dominant
attractive force between them. The van der Waals interfaces provide a novel platform for
exploring various fundamental physical phenomena in light-matter interaction, electronphonon
coupling, lattice strain engineering, charge-transfer dynamics and so on. New
functionalities can also be engineered in homogeneous/heterogeneous interfaces, which is
further enriched by the interlayer hybridization of electron wavefunctions. This has led
to realization of many rich and novel electronic phases like Mott-insulator and superconductivity.
The layer-hybridization and inter-layer electron-phonon interaction directly
determine in electrical transport across the van der Waals interface through coupling of
electrons to the interlayer breathing modes, but their impact on the thermal and thermoelectric
properties remain unexplored. The motivation of this thesis is to understand the
physical mechanism for thermoelectric transport across the sub-nanometer gap created at
the van der Waals interface of two layers of graphene.
We form atomically clean van der Waals interface at the junction between two independently
contacted graphene layers, called twisted bilayer graphene, in a eld-e ect geometry.
The crystallographic orientation of the participating layers was varied to tune the
interlayer electronic hybridization. Independent electrical contacts allowed us to investigate
both electrical and thermoelectric transport across the van der Waals junction as
function of doping.
To obtain thermopower or Seebeck coe cient (S) of the junction, we have employed Joule
heating in one of the graphene layers using sinusoidal current and the 2nd harmonic voltage
is measured between the two layers. The temperature di erence ( T) is measured
graphene in-plane resistance thermometry. We show that for large twist angle stacking,
i.e. lattice mis-orientation angle larger than about 4 degrees, the cross-plane Seebeck
coe cient, which is the ratio of 2nd harmonic voltage and T is driven by an e ective
interlayer phonon drag. The cross-plane thermo-voltage, which shows non-monotonic
behaviour with respect to both temperature and number density, is originated through
scattering of charge carriers by the out-of-plane layer breathing (ZO0/ZA2) phonon modes.
The resulting Seebeck coe cient shows signi cant deviation from the expected Landauer-
Buttiker formalism in the context of coherent transport in conventional tunnel junctions.
At small twist angle, however, interlayer hybridization of electron wavefunctions comes
into play and vertical transport is driven by momentum conserving coherent tunnelling.
We show that in presence of strong interlayer coupling the thermoelectric transport can be
described by the semiclassical Mott relation. Finally, exploiting the decoupling of charge
and heat at large lattice mis-orientation, we estimate that it is possible to achieve thermoelectric
gure-of-merit, or the ZT factor ( S2T= ) as large as 1 at room temperature,
surpassing most common bulk thermoelectric materials around room temperature.
In summary, the contrasting nature of the thermoelectric transport for small and large
rotational stacking provides tunability of coherence motion of charge carriers through
atomically-layered hybrids which can be manifested in engineering new phases of thermoelectricity
in van der Waals epitaxy
Transport, localization and entanglement in disordered and interacting systems: From real space to Fock space
No full textIn this thesis, we explore some of the exciting physics of condensed matter systems
manifested because of imperfection or disorder and interactions among the constituent
particles. In phenomena like transport, e.g., electrical current; localization, e.g.,
confinement of electrons only within a small part of a system; entanglement (a
correlation among the constituents particle); disorder and interaction play essential
roles. These three properties are our main focus in the thesis.
There are six chapters. In the first chapter, we introduce a few landmarks in the
field to set the stage and give an overview of the works presented in the thesis. In the
second chapter, we consider quasi-disordered or quasiperiodic systems in one, two, and
three dimensions, where the quasi-disorder is deterministic but non-repeating
throughout a lattice and considered from. Metal-insulator transitions in these systems
are probed by calculating conductances and their change with system size. More
specifically, we look at the systems from the perspective of single-parameter scaling
theory. In the third chapter, we consider both the disordered and quasi-disordered
systems with interactions. The systems show transitions from thermal to many-body
localized phases, and we study them in Fock space, which is a natural description for an
interacting system. We exploit the Fock space structure to calculate the propagator or
Green’s function in an iterative way to push the system size accessible in the exact
calculations. We define a length scale in Fock space, which can detect the phase transition
and distinguish between the disordered and the quasi-disordered systems. In the fourth
chapter, motivated by an experiment, we study the electrical current and noise therein in
a disordered quantum Hall system in the proximity of a superconductor. To our surprise,
the quantum Hall conductance plateau in the system comes with noise in the current as
also observed in the experiment, and the calculated quantities match pretty well with the
observed values. In the fifth chapter, we study the entanglement entropy of an interacting
fermionic system using a new saddle-point approximation similar to a mean-field
approximation. The approximation is based on a newly developed path integral approach
for calculating the entanglement entropy. In the last chapter, we conclude the thesis by
summarizing the important findings of our works presented in the thesis with some
future directions
PENGARUH KOMUNIKASI PEMASARAN TERHADAP KEPUTUSAN MEMBELI PADA HERO SUPERMARKET GATOT SUBROTO
No full textPUSPITA LIANTI PUTRI. 2009. The Impact of Marketing Communication to Purchase Decision at Hero Supermarket Gatot Subroto. DIII Marketing Management Study Program. Departement of Management. Faculty of Economics. Satate University of Jakarta.
Scientific work aims to determine the impact of marketing communications to purchase decision at Hero Supermarket Gatot Subroto. The author takes the title is to know about marketing communication activities conducted Hero Supermarket Gatot Subroto and find out whether or not the impact of marketing communications to purchase decision at Hero Supermarket Gatot Subroto. The method used in this research is descriptive analysis by using data gathered through a literature study, interviews and questionnaires. Type of scale used is a Likert scale. The sampling technique used by random sampling. In this study the authors used the technique of data analysis qualitative and quantitative with product moment correlation.
From the result of this scientific work we can conclude that the presence of a positive relationship between marketing communication with purchase decision at Hero Supermarket Gatot Subroto is 0.55 and the magnitude of the impact of marketing communications in Hero Supermarket Gatot Subroto explained the purchase decision is 30.25% with a significance level 5% ( = 5%). Thus we can conclude that H0 is rejected and H1 is accepted
Criterion for the occurrence of many-body localization in the presence of a single-particle mobility edge
No full textNoninteracting fermions in one dimension can undergo a localization-delocalization transition in the presence of a quasiperiodic potential as a function of that potential. In the presence of interactions, this transition transforms into a many-body localization (MBL) transition. Recent studies have suggested that this type of transition can also occur in models with quasiperiodic potentials that possess single-particle mobility edges. Two such models were studied by Modak and Mukerjee Phys. Rev. Lett. 115, 230401 (2015)] but only one was found to exhibit an MBL transition in the presence of interactions while the other one did not. In this work we investigate the occurrence of MBL in the presence of weak interactions in five different models with single-particle mobility edges in one dimension with a view to obtaining a criterion for the same. We find that not all such models undergo a thermal-MBL phase transition in the presence of weak interactions. We propose a criterion to determine whether MBL is likely to occur in the presence of interaction based only on the properties of the noninteracting models. The relevant quantity epsilon is a measure of how localized the localized states are relative to how delocalized the delocalized states are in the noninteracting model. We also study various other features of the noninteracting models such as the divergence of the localization length at the mobility edge and the presence or absence of ``ergodicity'' and localization in their many-body eigenstates. However, we find that these features cannot be used to predict the occurrence of MBL upon the introduction of weak interactions
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